NYLL
Module

RationalCyclotomicFinitePlaceArtin.lean

Nyll.ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.RationalCyclotomicFinitePlaceArtin
156 declarations indexed from sourceView source ↗

Module documentation

Finite-place Artin symbols in rational cyclotomic levels

At a rational prime away from the cyclotomic level, the chosen completed extension is unramified. Its normalized local Artin map is therefore the arithmetic Frobenius raised to the local valuation. The genuine primitive root in the localized cyclotomic level identifies the image of arithmetic Frobenius under the global cyclotomic character with the residue prime.

Named compatibility witnesses used by the ramified-prime and ray-norm modules. They are not installed as a duplicate module-level instance family; the canonical instances above already provide the same data.

Ramified prime-power transport

This section reuses the canonical finite-place Artin tower above. In particular, it introduces no parallel completion/localization instance tower.

The ramified principal finite-place factor

Imports23
Imported by3

Declarations

instance

rationalFinitePlaceCompletionRatAlgebra

L54 ↗
GlobalClassFieldTheory.Reciprocity.rationalFinitePlaceCompletionRatAlgebra
Source context 5 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
Lean source · statement
instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion
theorem

mappedAbelianLocalArtin_eq_frobenius_zpow

L78 ↗
GlobalClassFieldTheory.Reciprocity.mappedAbelianLocalArtin_eq_frobenius_zpow
Source context 13 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
Lean source · statement
theorem mappedAbelianLocalArtin_eq_frobenius_zpow
    {F E G : Type}
    [Field F] [ValuativeRel F] [TopologicalSpace F]
    [IsNonarchimedeanLocalField F]
    [Field E] [ValuativeRel E] [UniformSpace E] [IsUniformAddGroup E]
    [IsNonarchimedeanLocalField E]
    [Algebra F E] [FiniteDimensional F E] [IsAbelianGalois F E]
    [Valuation.HasExtension
      (ValuativeRel.valuation F) (ValuativeRel.valuation E)]
    [IsNonarchimedeanLocalField.IsUnramifiedValuedExtension F E]
    [Group G]
    (f : (E ≃ₐ[F] E) →* G) (x : Fˣ) :
    f (LocalClassFieldTheory.abelianLocalArtinMonoidHom F E x) =
      (f (arithmeticFrobeniusOfUnramifiedValuation F E)) ^
        IsNonarchimedeanLocalField.valuationMap F
          (Additive.ofMul x)
instance

(anonymous instance)

L97 ↗
Source context 13 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
Lean source · statement
instance (q : Nat.Primes) : Fact q.1.Prime
instance

(anonymous instance)

L100 ↗
Source context 14 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
Lean source · statement
instance (m : ℕ+) : NeZero (m : ℕ)
instance

rationalCyclotomicLevelFiniteDimensional

L103 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicLevelFiniteDimensional
Source context 15 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
Lean source · statement
instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m)
instance

rationalCyclotomicLevelIsAbelianGalois

L111 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicLevelIsAbelianGalois
Source context 16 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
Lean source · statement
instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m)
instance

rationalFinitePlaceBaseNontriviallyNormedField

L129 ↗
GlobalClassFieldTheory.Reciprocity.rationalFinitePlaceBaseNontriviallyNormedField
Source context 17 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
Lean source · statement
instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion
instance

rationalFinitePlaceBaseLocallyCompactSpace

L140 ↗
GlobalClassFieldTheory.Reciprocity.rationalFinitePlaceBaseLocallyCompactSpace
Source context 18 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
Lean source · statement
instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion
instance

rationalFinitePlaceBaseIsUltrametricDist

L149 ↗
GlobalClassFieldTheory.Reciprocity.rationalFinitePlaceBaseIsUltrametricDist
Source context 19 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
Lean source · statement
instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion
instance

rationalFinitePlaceBaseValued

L161 ↗
GlobalClassFieldTheory.Reciprocity.rationalFinitePlaceBaseValued
Source context 20 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
Lean source · statement
instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0
instance

rationalFinitePlaceBaseValuativeRel

L173 ↗
GlobalClassFieldTheory.Reciprocity.rationalFinitePlaceBaseValuativeRel
Source context 21 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
Lean source · statement
instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion
instance

rationalFinitePlaceBaseValuationIsNontrivial

L181 ↗
GlobalClassFieldTheory.Reciprocity.rationalFinitePlaceBaseValuationIsNontrivial
Source context 22 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial
instance

rationalFinitePlaceBaseValuationCompatible

L195 ↗
GlobalClassFieldTheory.Reciprocity.rationalFinitePlaceBaseValuationCompatible
Source context 23 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
Lean source · statement
instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible
instance

rationalFinitePlaceBaseValuativeRelIsNontrivial

L204 ↗
GlobalClassFieldTheory.Reciprocity.rationalFinitePlaceBaseValuativeRelIsNontrivial
Source context 24 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
Lean source · statement
instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion
instance

rationalFinitePlaceBaseIsValuativeTopology

L217 ↗
GlobalClassFieldTheory.Reciprocity.rationalFinitePlaceBaseIsValuativeTopology
Source context 25 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
Lean source · statement
instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion
instance

rationalFinitePlaceBaseIsNonarchimedeanLocalField

L226 ↗
GlobalClassFieldTheory.Reciprocity.rationalFinitePlaceBaseIsNonarchimedeanLocalField
Source context 26 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
Lean source · statement
instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion
theorem

rationalCyclotomicPrincipalPrimeLevelFiniteDimensional

L240 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicPrincipalPrimeLevelFiniteDimensional
Documentation

Rational cyclotomic levels are finite-dimensional over ℚ.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
theorem rationalCyclotomicPrincipalPrimeLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ (KummerTheory.rationalCyclotomicLevel m)
theorem

rationalCyclotomicPrincipalPrimeLevelIsAbelianGalois

L246 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicPrincipalPrimeLevelIsAbelianGalois
Documentation

Rational cyclotomic levels are abelian Galois extensions of ℚ.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
theorem rationalCyclotomicPrincipalPrimeLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ (KummerTheory.rationalCyclotomicLevel m)
def

rationalPrimeFactorCompletionNontriviallyNormedField

L254 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrimeFactorCompletionNontriviallyNormedField
Documentation

The canonical nontrivially normed field structure on the completion of ℚ at the rational prime p, exposed for principal-prime constructions.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
def rationalPrimeFactorCompletionNontriviallyNormedField
    (p : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime p)).Completion
theorem

rationalPrimeFactorCompletionLocallyCompactSpace

L262 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrimeFactorCompletionLocallyCompactSpace
Documentation

The completion of ℚ at p is locally compact.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
theorem rationalPrimeFactorCompletionLocallyCompactSpace
    (p : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime p)).Completion
theorem

rationalPrimeFactorCompletionIsUltrametricDist

L270 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrimeFactorCompletionIsUltrametricDist
Documentation

The completion of ℚ at p carries its canonical ultrametric distance.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
theorem rationalPrimeFactorCompletionIsUltrametricDist
    (p : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime p)).Completion
def

rationalPrimeFactorCompletionValued

L279 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrimeFactorCompletionValued
Documentation

The canonical ℝ≥0-valued structure on the completion of ℚ at p.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
def rationalPrimeFactorCompletionValued
    (p : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime p)).Completion ℝ≥0
def

rationalPrimeFactorCompletionValuativeRel

L289 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrimeFactorCompletionValuativeRel
Documentation

The valuative relation induced by the canonical valuation on the completion of ℚ at p.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
def rationalPrimeFactorCompletionValuativeRel
    (p : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime p)).Completion
theorem

rationalPrimeFactorCompletionValuationIsNontrivial

L297 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrimeFactorCompletionValuationIsNontrivial
Documentation

The canonical valuation on the completion of ℚ at p is nontrivial.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
theorem rationalPrimeFactorCompletionValuationIsNontrivial
    (p : Nat.Primes) :
    (Valued.v : Valuation
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime p)).Completion ℝ≥0).IsNontrivial
theorem

rationalPrimeFactorCompletionValuationCompatible

L306 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrimeFactorCompletionValuationCompatible
Documentation

The canonical valuation on the completion of ℚ at p is compatible with its field structure.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
theorem rationalPrimeFactorCompletionValuationCompatible
    (p : Nat.Primes) :
    (Valued.v : Valuation
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime p)).Completion ℝ≥0).Compatible
theorem

rationalPrimeFactorCompletionValuativeRelIsNontrivial

L314 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrimeFactorCompletionValuativeRelIsNontrivial
Documentation

The canonical valuative relation on the completion at p is nontrivial.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
theorem rationalPrimeFactorCompletionValuativeRelIsNontrivial
    (p : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime p)).Completion
theorem

rationalPrimeFactorCompletionIsValuativeTopology

L322 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrimeFactorCompletionIsValuativeTopology
Documentation

The completion topology at p is induced by its canonical valuation.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
theorem rationalPrimeFactorCompletionIsValuativeTopology
    (p : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime p)).Completion
theorem

rationalPrimeFactorCompletionIsNonarchimedeanLocalField

L330 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrimeFactorCompletionIsNonarchimedeanLocalField
Documentation

The completion of ℚ at p is a nonarchimedean local field.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
theorem rationalPrimeFactorCompletionIsNonarchimedeanLocalField
    (p : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime p)).Completion
def

rationalCyclotomicPrincipalPrimeModulus

L339 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicPrincipalPrimeModulus
Documentation

The positive conductor of the n-th ramified cyclotomic level at p.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
def rationalCyclotomicPrincipalPrimeModulus
    (p : Nat.Primes) (n : ℕ) : ℕ+
abbrev

RationalCyclotomicPrincipalPrimeCompletion

L344 ↗
GlobalClassFieldTheory.Reciprocity.RationalCyclotomicPrincipalPrimeCompletion
Documentation

The rational finite-place completion used at the prime p.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
abbrev RationalCyclotomicPrincipalPrimeCompletion
    (p : Nat.Primes)
abbrev

RationalCyclotomicPrincipalPrimeLocalizedLevel

L350 ↗
GlobalClassFieldTheory.Reciprocity.RationalCyclotomicPrincipalPrimeLocalizedLevel
Documentation

The chosen localized cyclotomic field at level p ^ (n + 1).

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
abbrev RationalCyclotomicPrincipalPrimeLocalizedLevel
    (p : Nat.Primes) (n : ℕ)
abbrev

RationalCyclotomicPrincipalPrimePadicLevel

L357 ↗
GlobalClassFieldTheory.Reciprocity.RationalCyclotomicPrincipalPrimePadicLevel
Documentation

The standard multiplicative Lubin--Tate field at level n.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
abbrev RationalCyclotomicPrincipalPrimePadicLevel
    (p : Nat.Primes) (n : ℕ)
def

rationalFinitePlaceCompletionIntegerRingEquivPadicInt

L364 ↗
GlobalClassFieldTheory.Reciprocity.rationalFinitePlaceCompletionIntegerRingEquivPadicInt
Documentation

The valuation ring of the absolute-value completion at q, identified with the standard p-adic integer ring ℤ_q.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
def rationalFinitePlaceCompletionIntegerRingEquivPadicInt
    (q : Nat.Primes) :
    𝒪[(HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion] ≃+*
      ℤ_[q.1]
def

rationalFinitePlaceCompletionRingEquivPadic

L383 ↗
GlobalClassFieldTheory.Reciprocity.rationalFinitePlaceCompletionRingEquivPadic
Documentation

The absolute-value completion at the rational prime q, identified with the standard field ℚ_q.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
def rationalFinitePlaceCompletionRingEquivPadic
    (q : Nat.Primes) :
    (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ≃+*
      ℚ_[q.1]
theorem

rationalFinitePlaceCompletionRingEquivPadic_algebraMap

L392 ↗
GlobalClassFieldTheory.Reciprocity.rationalFinitePlaceCompletionRingEquivPadic_algebraMap
Documentation

The completion-to-ℚ_q equivalence respects the rational embedding.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
theorem rationalFinitePlaceCompletionRingEquivPadic_algebraMap
    (q : Nat.Primes) (a : ℚ) :
    rationalFinitePlaceCompletionRingEquivPadic q
        (algebraMap ℚ
          (HeightOneSpectrum.adicAbv ℚ
            (RayClass.rationalPrime q)).Completion a) =
      algebraMap ℚ ℚ_[q.1] a
theorem

rationalFinitePlaceCompletionIntegerRingEquivPadicInt_coe

L404 ↗
GlobalClassFieldTheory.Reciprocity.rationalFinitePlaceCompletionIntegerRingEquivPadicInt_coe
Documentation

The completion field equivalence and its restriction to valuation rings commute with the natural inclusions into the fields.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
theorem rationalFinitePlaceCompletionIntegerRingEquivPadicInt_coe
    (q : Nat.Primes)
    (a :
      𝒪[(HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion]) :
    rationalFinitePlaceCompletionRingEquivPadic q
        (algebraMap
          𝒪[(HeightOneSpectrum.adicAbv ℚ
            (RayClass.rationalPrime q)).Completion]
          (HeightOneSpectrum.adicAbv ℚ
            (RayClass.rationalPrime q)).Completion a) =
      algebraMap ℤ_[q.1] ℚ_[q.1]
        (rationalFinitePlaceCompletionIntegerRingEquivPadicInt q a)
theorem

rationalFinitePlaceCompletionRingEquivPadic_semilinearValuationCompatible

L425 ↗
GlobalClassFieldTheory.Reciprocity.rationalFinitePlaceCompletionRingEquivPadic_semilinearValuationCompatible
Documentation

The canonical rational-completion equivalence preserves the canonical valuations.

Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
open _root_.ValuationTheory.DiscreteValuationField.ValuedExtension
  (valuation_hasExtension_of_valuationSubring_equiv) in
Lean source · statement
theorem
    rationalFinitePlaceCompletionRingEquivPadic_semilinearValuationCompatible
    (p : Nat.Primes) :
    SemilinearValuationCompatible
      (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1]
      (rationalFinitePlaceCompletionRingEquivPadic p)
def

rationalPrimeFinitePlaceInteger

L495 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrimeFinitePlaceInteger
Documentation

The rational prime, pulled back from ℤ_q to the valuation ring of the absolute-value completion at q.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
def rationalPrimeFinitePlaceInteger
    (q : Nat.Primes) :
    𝒪[(HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion]
theorem

rationalPrimeFinitePlaceInteger_irreducible

L504 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrimeFinitePlaceInteger_irreducible
Documentation

The pulled-back rational prime is irreducible in the completion valuation ring.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
theorem rationalPrimeFinitePlaceInteger_irreducible
    (q : Nat.Primes) :
    Irreducible (rationalPrimeFinitePlaceInteger q)
theorem

rationalPrimeFinitePlaceInteger_coe

L516 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrimeFinitePlaceInteger_coe
Documentation

Coercing the pulled-back prime to the completion field gives the ordinary image of the rational number q.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
theorem rationalPrimeFinitePlaceInteger_coe
    (q : Nat.Primes) :
    ((rationalPrimeFinitePlaceInteger q :
      𝒪[(HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion]) :
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion) =
      algebraMap ℚ
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        (q.1 : ℚ)
def

rationalPrimeFinitePlaceFieldUnit

L572 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrimeFinitePlaceFieldUnit
Documentation

The rational prime as a field unit of its absolute-value completion.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
def rationalPrimeFinitePlaceFieldUnit
    (q : Nat.Primes) :
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completionˣ
theorem

rationalPrimeFinitePlaceFieldUnit_valuationMap

L588 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrimeFinitePlaceFieldUnit_valuationMap
Documentation

In the inverse-standard local reciprocity normalization, the rational prime itself has normalized additive value -1.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
theorem rationalPrimeFinitePlaceFieldUnit_valuationMap
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField.valuationMap
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        (Additive.ofMul
          (rationalPrimeFinitePlaceFieldUnit q)) =
      -1
def

rationalPrimeUnitFinitePlaceIntegerUnit

L606 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrimeUnitFinitePlaceIntegerUnit
Documentation

The rational q-unit part of x, pulled back from ℤ_qˣ to the valuation ring of the absolute-value completion.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
def rationalPrimeUnitFinitePlaceIntegerUnit
    (x : ℚˣ) (q : Nat.Primes) :
    𝒪[(HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion]ˣ
theorem

rationalPrimeUnitFinitePlaceIntegerUnit_coe

L620 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrimeUnitFinitePlaceIntegerUnit_coe
Documentation

Forgetting the integrality proof from the pulled-back q-unit gives the ordinary image of the rational q-unit in the completion field.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
theorem rationalPrimeUnitFinitePlaceIntegerUnit_coe
    (x : ℚˣ) (q : Nat.Primes) :
    (((rationalPrimeUnitFinitePlaceIntegerUnit x q :
        𝒪[(HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion]ˣ) :
        𝒪[(HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion]) :
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion) =
      algebraMap ℚ
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        (rationalPrimeUnit x q : ℚ)
theorem

rationalPrimeUnitFinitePlaceField_valuationMap

L691 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrimeUnitFinitePlaceField_valuationMap
Documentation

The completion field unit underlying the pulled-back rational q-unit has normalized additive value zero.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
theorem rationalPrimeUnitFinitePlaceField_valuationMap
    (x : ℚˣ) (q : Nat.Primes) :
    IsNonarchimedeanLocalField.valuationMap
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        (Additive.ofMul
          (IsNonarchimedeanLocalField.integerUnitsToFieldUnits
            (HeightOneSpectrum.adicAbv ℚ
              (RayClass.rationalPrime q)).Completion
            (rationalPrimeUnitFinitePlaceIntegerUnit x q))) =
      0
def

rationalPrimeGeneratorUnit

L710 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrimeGeneratorUnit
Documentation

The rational prime q, regarded as a unit of ℚ.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
def rationalPrimeGeneratorUnit (q : Nat.Primes) : ℚˣ
theorem

rationalPrimeGeneratorUnit_coe

L715 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrimeGeneratorUnit_coe
Documentation

The underlying rational number of the prime generator unit is q.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
theorem rationalPrimeGeneratorUnit_coe (q : Nat.Primes) :
    (rationalPrimeGeneratorUnit q : ℚ) = q.1
theorem

rationalPrimeGeneratorUnit_zpow_mul_rationalPrimeUnit

L721 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrimeGeneratorUnit_zpow_mul_rationalPrimeUnit
Documentation

Reattaching the removed q-power to the rational q-unit recovers the original rational field unit.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
theorem rationalPrimeGeneratorUnit_zpow_mul_rationalPrimeUnit
    (x : ℚˣ) (q : Nat.Primes) :
    rationalPrimeGeneratorUnit q ^
          padicValRat q.1 (x : ℚ) *
        rationalPrimeUnit x q =
      x
def

rationalPrincipalFinitePlaceInput

L733 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrincipalFinitePlaceInput
Documentation

The source unit in the absolute-value completion represented by the finite component of a rational principal idele.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
def rationalPrincipalFinitePlaceInput
    (x : ℚˣ) (q : Nat.Primes) :
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completionˣ
theorem

rationalPrincipalFinitePlaceInput_eq_algebraMap

L746 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrincipalFinitePlaceInput_eq_algebraMap
Documentation

The source unit represented by a principal finite component is the ordinary image of the rational field unit in the absolute-value completion.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
theorem rationalPrincipalFinitePlaceInput_eq_algebraMap
    (x : ℚˣ) (q : Nat.Primes) :
    rationalPrincipalFinitePlaceInput x q =
      Units.map
        (algebraMap ℚ
          (HeightOneSpectrum.adicAbv ℚ
            (RayClass.rationalPrime q)).Completion).toMonoidHom
        x
theorem

rationalPrincipalFiniteComponent_valuationMap

L812 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrincipalFiniteComponent_valuationMap
Documentation

The normalized local exponent of a rational principal finite component is the negative of the usual q-adic exponent. The minus sign records the inverse-standard local reciprocity convention in which a prime element has normalized value -1.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
theorem rationalPrincipalFiniteComponent_valuationMap
    (x : ℚˣ) (q : Nat.Primes) :
    IsNonarchimedeanLocalField.valuationMap
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        (Additive.ofMul
          ((finitePlaceCompletionUnitsContinuousMulEquiv
            (RayClass.rationalPrime q)).symm
            (IdeleGroup.finiteComponent
              (RayClass.rationalPrime q)
              (IdeleGroup.principalIdele ℚ x)))) =
      -padicValRat q.1 (x : ℚ)
theorem

rationalPrimePrincipalFiniteComponent_valuationMap

L871 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrimePrincipalFiniteComponent_valuationMap
Documentation

The principal finite component of the rational prime itself has normalized local exponent -1.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
theorem rationalPrimePrincipalFiniteComponent_valuationMap
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField.valuationMap
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        (Additive.ofMul
          ((finitePlaceCompletionUnitsContinuousMulEquiv
            (RayClass.rationalPrime q)).symm
            (IdeleGroup.finiteComponent
              (RayClass.rationalPrime q)
              (IdeleGroup.principalIdele ℚ
                (rationalPrimeGeneratorUnit q))))) =
      -1
theorem

rationalCyclotomicLevel_galEquivZMod_eq_unitOfCoprime

L890 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicLevel_galEquivZMod_eq_unitOfCoprime
Documentation

A cyclotomic automorphism which raises the selected primitive root to the q-th power has cyclotomic character equal to the residue-prime unit.

Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
theorem rationalCyclotomicLevel_galEquivZMod_eq_unitOfCoprime
    (m : ℕ+) (q : Nat.Primes)
    (hq : ¬ q.1 ∣ (m : ℕ))
    (σ : KummerTheory.rationalCyclotomicLevel m ≃ₐ[ℚ]
      KummerTheory.rationalCyclotomicLevel m)
    (hσ :
      σ (rationalCyclotomicLevelPrimitiveRoot m) =
        rationalCyclotomicLevelPrimitiveRoot m ^ q.1) :
    IsCyclotomicExtension.Rat.galEquivZMod
        (m : ℕ) (KummerTheory.rationalCyclotomicLevel m) σ =
      ZMod.unitOfCoprime q.1
        (q.2.coprime_iff_not_dvd.mpr hq)
abbrev

rationalCyclotomicArtinPlace

L926 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinPlace
Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
abbrev rationalCyclotomicArtinPlace (q : Nat.Primes) :
    HeightOneSpectrum (𝓞 ℚ)
abbrev

rationalCyclotomicArtinBaseAbv

L930 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinBaseAbv
Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
abbrev rationalCyclotomicArtinBaseAbv (q : Nat.Primes) :
    AbsoluteValue ℚ ℝ
abbrev

rationalCyclotomicArtinLevel

L934 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLevel
Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
abbrev rationalCyclotomicArtinLevel (m : ℕ+)
abbrev

rationalCyclotomicArtinExtension

L937 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinExtension
Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
abbrev rationalCyclotomicArtinExtension
    (m : ℕ+) (q : Nat.Primes) :
    AbsoluteValueExtension
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinLevel m)
abbrev

rationalCyclotomicArtinLocalizedField

L946 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedField
Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
abbrev rationalCyclotomicArtinLocalizedField
    (m : ℕ+) (q : Nat.Primes)
instance

rationalCyclotomicArtinExtensionAlgebra

L953 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinExtensionAlgebra
Source context 27 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
Lean source · statement
instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion
instance

rationalCyclotomicArtinExtensionSMul

L961 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinExtensionSMul
Source context 28 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
Lean source · statement
instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion
instance

rationalCyclotomicArtinCompletionAlgebra

L968 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinCompletionAlgebra
Source context 29 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
Lean source · statement
instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion
instance

rationalCyclotomicArtinLocalizedAlgebra

L979 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedAlgebra
Source context 30 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
Lean source · statement
instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q)
instance

rationalCyclotomicArtinLocalizedGlobalAlgebra

L988 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedGlobalAlgebra
Source context 31 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
Lean source · statement
instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q)
instance

rationalCyclotomicArtinLocalizedGlobalSMul

L996 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedGlobalSMul
Source context 32 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
Lean source · statement
instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q)
instance

rationalCyclotomicArtinLocalizedScalarTower

L1002 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedScalarTower
Source context 33 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
Lean source · statement
instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q)
instance

rationalCyclotomicArtinLocalizedFiniteDimensional

L1013 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedFiniteDimensional
Source context 34 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
Lean source · statement
instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q)
instance

rationalCyclotomicArtinLocalizedIsAbelianGalois

L1024 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedIsAbelianGalois
Source context 35 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
Lean source · statement
instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q)
instance

rationalCyclotomicArtinLocalizedIsSeparable

L1037 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedIsSeparable
Source context 36 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
Lean source · statement
instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q)
instance

rationalCyclotomicArtinLocalizedIsCyclotomic

L1045 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedIsCyclotomic
Source context 37 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
Lean source · statement
instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q)
instance

rationalCyclotomicArtinExtensionFiniteDimensional

L1054 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinExtensionFiniteDimensional
Source context 38 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
Lean source · statement
instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion
instance

rationalCyclotomicArtinExtensionContinuousSMul

L1066 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinExtensionContinuousSMul
Source context 39 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
Lean source · statement
instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion
instance

rationalCyclotomicArtinExtensionLocallyCompact

L1078 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinExtensionLocallyCompact
Source context 40 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
Lean source · statement
instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion
def

rationalCyclotomicArtinLocalizedEquivCompletion

L1087 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedEquivCompletion
Source context 41 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
Lean source · statement
def
    rationalCyclotomicArtinLocalizedEquivCompletion
    (m : ℕ+) (q : Nat.Primes) :
    rationalCyclotomicArtinLocalizedField m q ≃ᵢ
      (rationalCyclotomicArtinExtension m q).1.Completion
instance

rationalCyclotomicArtinLocalizedLocallyCompact

L1101 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedLocallyCompact
Source context 41 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
Lean source · statement
instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q)
instance

rationalCyclotomicArtinLocalizedIsUltrametricDist

L1109 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedIsUltrametricDist
Source context 42 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
Lean source · statement
instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q)
instance

rationalCyclotomicArtinLocalizedValued

L1120 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedValued
Source context 43 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
Lean source · statement
instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
instance

rationalCyclotomicArtinLocalizedValuativeRel

L1130 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedValuativeRel
Source context 44 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
Lean source · statement
instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q)
instance

rationalCyclotomicArtinLocalizedValuationCompatible

L1140 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedValuationCompatible
Source context 45 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
Lean source · statement
instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible
instance

rationalCyclotomicArtinLocalizedValuationHasExtension

L1147 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedValuationHasExtension
Source context 46 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
Lean source · statement
instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q))
instance

rationalCyclotomicArtinLocalizedValuationIsNontrivial

L1161 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedValuationIsNontrivial
Source context 47 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
Lean source · statement
instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial
instance

rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial

L1172 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
Source context 48 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
Lean source · statement
instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q)
instance

rationalCyclotomicArtinLocalizedIsValuativeTopology

L1181 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedIsValuativeTopology
Source context 49 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
Lean source · statement
instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q)
instance

rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField

L1189 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
Source context 50 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
Lean source · statement
instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q)
instance

rationalCyclotomicArtinLocalizedIntegerAlgebra

L1198 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedIntegerAlgebra
Source context 51 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
Lean source · statement
instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q)
instance

rationalCyclotomicArtinLocalizedIsIntegralClosure

L1207 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedIsIntegralClosure
Source context 52 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
Lean source · statement
instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q)
instance

rationalCyclotomicArtinLocalizedIntegerModuleFinite

L1222 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedIntegerModuleFinite
Source context 53 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
Lean source · statement
instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q]
instance

rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension

L1239 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
Source context 55 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
Lean source · statement
instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
abbrev

rationalCyclotomicPrincipalPrimePlace

L1254 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicPrincipalPrimePlace
Source context 56 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
Lean source · statement
abbrev rationalCyclotomicPrincipalPrimePlace
    (p : Nat.Primes) : HeightOneSpectrum (𝓞 ℚ)
abbrev

rationalCyclotomicPrincipalPrimeLevel

L1258 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicPrincipalPrimeLevel
Source context 56 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
Lean source · statement
abbrev rationalCyclotomicPrincipalPrimeLevel
    (m : ℕ+)
abbrev

rationalCyclotomicPrincipalPrimeExtension

L1262 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicPrincipalPrimeExtension
Source context 56 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
Lean source · statement
abbrev rationalCyclotomicPrincipalPrimeExtension
    (m : ℕ+) (p : Nat.Primes)
def

rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra

L1269 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra
Documentation

The ℚ_[p]-algebra structure on the localized cyclotomic completion, transported through the canonical comparison with the p-adic completion.

Source context 56 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
Lean source · statement
def
    rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1]
      (rationalCyclotomicLocalizedCompletion m
        (RayClass.rationalPrime p))
instance

rationalCyclotomicArtinLocalizedPadicAlgebra

L1283 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedPadicAlgebra
Source context 56 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
Lean source · statement
instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p)
def

rationalFinitePlaceCompletionAlgEquivPadic

L1289 ↗
GlobalClassFieldTheory.Reciprocity.rationalFinitePlaceCompletionAlgEquivPadic
Source context 57 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
Lean source · statement
def rationalFinitePlaceCompletionAlgEquivPadic
    (p : Nat.Primes) :
    (rationalCyclotomicArtinBaseAbv p).Completion ≃ₐ[ℚ] ℚ_[p.1]
instance

rationalCyclotomicArtinLocalizedPadicScalarTower

L1296 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedPadicScalarTower
Source context 57 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
Lean source · statement
instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p)
theorem

rationalCyclotomicArtin_padic_algebraMap

L1306 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtin_padic_algebraMap
Source context 58 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
Lean source · statement
theorem rationalCyclotomicArtin_padic_algebraMap
    (m : ℕ+) (p : Nat.Primes) :
    algebraMap (rationalCyclotomicArtinBaseAbv p).Completion
        (rationalCyclotomicArtinLocalizedField m p) =
      (algebraMap ℚ_[p.1]
        (rationalCyclotomicArtinLocalizedField m p)) ∘
        (rationalFinitePlaceCompletionAlgEquivPadic p)
theorem

rationalCyclotomicArtin_algebraAdjoin_restrictScalars

L1327 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtin_algebraAdjoin_restrictScalars
Source context 58 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
Lean source · statement
theorem rationalCyclotomicArtin_algebraAdjoin_restrictScalars
    (m : ℕ+) (p : Nat.Primes) :
    (Algebra.adjoin (rationalCyclotomicArtinBaseAbv p).Completion
      (Set.singleton (rationalCyclotomicLocalizedPrimitiveRoot m
        (rationalCyclotomicArtinPlace p)) :
        Set (rationalCyclotomicArtinLocalizedField m p))).restrictScalars ℚ =
      (Algebra.adjoin ℚ_[p.1]
        (Set.singleton (rationalCyclotomicLocalizedPrimitiveRoot m
          (rationalCyclotomicArtinPlace p)) :
          Set (rationalCyclotomicArtinLocalizedField m p))).restrictScalars ℚ
theorem

rationalCyclotomicArtin_baseAlgebraAdjoin_restrict_eq_top

L1346 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtin_baseAlgebraAdjoin_restrict_eq_top
Source context 58 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
Lean source · statement
theorem
    rationalCyclotomicArtin_baseAlgebraAdjoin_restrict_eq_top
    (m : ℕ+) (p : Nat.Primes) :
    (Algebra.adjoin (rationalCyclotomicArtinBaseAbv p).Completion
      (Set.singleton (rationalCyclotomicLocalizedPrimitiveRoot m
        (rationalCyclotomicArtinPlace p)) :
        Set (rationalCyclotomicArtinLocalizedField m p))).restrictScalars ℚ =
      (⊤ : Subalgebra (rationalCyclotomicArtinBaseAbv p).Completion
      (rationalCyclotomicArtinLocalizedField m p)).restrictScalars ℚ
theorem

rationalCyclotomicArtin_restrictScalars_top_base_eq_padic

L1375 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtin_restrictScalars_top_base_eq_padic
Source context 58 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
Lean source · statement
theorem rationalCyclotomicArtin_restrictScalars_top_base_eq_padic
    (m : ℕ+) (p : Nat.Primes) :
    (⊤ : Subalgebra (rationalCyclotomicArtinBaseAbv p).Completion
        (rationalCyclotomicArtinLocalizedField m p)).restrictScalars ℚ =
      (⊤ : Subalgebra ℚ_[p.1]
        (rationalCyclotomicArtinLocalizedField m p)).restrictScalars ℚ
theorem

rationalCyclotomicArtin_padicAlgebraAdjoin_restrict_eq_top

L1384 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtin_padicAlgebraAdjoin_restrict_eq_top
Source context 58 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
Lean source · statement
theorem
    rationalCyclotomicArtin_padicAlgebraAdjoin_restrict_eq_top
    (m : ℕ+) (p : Nat.Primes) :
    (Algebra.adjoin ℚ_[p.1]
      (Set.singleton (rationalCyclotomicLocalizedPrimitiveRoot m
        (rationalCyclotomicArtinPlace p)) :
        Set (rationalCyclotomicArtinLocalizedField m p))).restrictScalars ℚ =
      (⊤ : Subalgebra ℚ_[p.1]
        (rationalCyclotomicArtinLocalizedField m p)).restrictScalars ℚ
theorem

rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional

L1400 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
Documentation

The finite-dimensional instance for the standard multiplicative level, named once so all consumers use the same proof term.

Source context 58 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
Lean source · statement
theorem rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
    (p : Nat.Primes) (n : ℕ) :
    FiniteDimensional ℚ_[p.1]
      (standardLubinTateLevelField
        (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n)
instance

rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois

L1411 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
Source context 59 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
Lean source · statement
instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n)
instance

rationalPrimeFactorCompletionPadicAlgebra

L1421 ↗
GlobalClassFieldTheory.Reciprocity.rationalPrimeFactorCompletionPadicAlgebra
Source context 60 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
Lean source · statement
instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1]
theorem

padicMultiplicativePrimitiveRoot_adjoin_eq_top

L1428 ↗
GlobalClassFieldTheory.Reciprocity.padicMultiplicativePrimitiveRoot_adjoin_eq_top
Documentation

The genuine multiplicative Lubin--Tate level is generated by its primitive p ^ (n + 1)-st root of unity.

Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · full declaration
theorem padicMultiplicativePrimitiveRoot_adjoin_eq_top
    (p : ℕ) [Fact p.Prime] (n : ℕ) :
    let hπ := padicMultiplicativeLubinTateSeries_isUniformizer p
    let T := standardLubinTateLevelField hπ n
    Algebra.adjoin ℚ_[p]
        ({padicMultiplicativePrimitiveRoot p n} : Set T) =
      ⊤ := by
  let hπ := padicMultiplicativeLubinTateSeries_isUniformizer p
  let T := standardLubinTateLevelField hπ n
  let ζ : T := padicMultiplicativePrimitiveRoot p n
  let m := p ^ (n + 1)
  let : NeZero m :=
    ⟨pow_ne_zero _ (Fact.out : Nat.Prime p).ne_zero⟩
  let : FiniteDimensional ℚ_[p] T :=
    standardLubinTateLevelField_finiteDimensional hπ n
  have hζ : IsPrimitiveRoot ζ m := by
    simpa only [ζ, m] using
      padicMultiplicativePrimitiveRoot_isPrimitiveRoot p n
  let A : IntermediateField ℚ_[p] T :=
    IntermediateField.adjoin ℚ_[p] {ζ}
  let : IsCyclotomicExtension {m} ℚ_[p] A :=
    hζ.intermediateField_adjoin_isCyclotomicExtension ℚ_[p]
  have hAfin :
      Module.finrank ℚ_[p] A = Nat.totient m := by
    exact
      IsCyclotomicExtension.finrank A
        (by
          simpa only [m] using
            padicCyclotomicPolynomial_irreducible_prime_pow_succ
              p n)
  have hTfin :
      Module.finrank ℚ_[p] T = Nat.totient m := by
    rw [standardLubinTateLevelField_finrank hπ n]
    have hcard :
        Nat.card (padicLocalField p).residueField = p := by
      simpa [padicLocalField] using
        padicCompleteDVF_residueField_card p
    rw [hcard, Nat.totient_prime_pow
      (Fact.out : Nat.Prime p) (Nat.succ_pos n)]
    simp [Nat.mul_comm]
  have hAeq : A = ⊤ := by
    apply IntermediateField.eq_of_le_of_finrank_eq le_top
    simpa using hAfin.trans hTfin.symm
  calc
    Algebra.adjoin ℚ_[p] {ζ} = A.toSubalgebra := by
      exact
        (IntermediateField.adjoin_toSubalgebra
          ({ζ} : Set T)).symm
    _ = (⊤ : IntermediateField ℚ_[p] T).toSubalgebra :=
      congrArg IntermediateField.toSubalgebra hAeq
    _ = ⊤ := rfl
theorem

padicMultiplicativeLevel_isCyclotomicExtension

L1482 ↗
GlobalClassFieldTheory.Reciprocity.padicMultiplicativeLevel_isCyclotomicExtension
Documentation

The standard multiplicative Lubin--Tate level is the actual p ^ (n + 1)-cyclotomic extension of ℚ_p.

Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · full declaration
theorem padicMultiplicativeLevel_isCyclotomicExtension
    (p : ℕ) [Fact p.Prime] (n : ℕ) :
    let hπ := padicMultiplicativeLubinTateSeries_isUniformizer p
    let T := standardLubinTateLevelField hπ n
    IsCyclotomicExtension {p ^ (n + 1)} ℚ_[p] T := by
  let hπ := padicMultiplicativeLubinTateSeries_isUniformizer p
  let T := standardLubinTateLevelField hπ n
  exact
    padic_isCyclotomicExtension_of_primitiveRoot_adjoin_eq_top
      (padicMultiplicativePrimitiveRoot p n)
      (padicMultiplicativePrimitiveRoot_isPrimitiveRoot p n)
      (padicMultiplicativePrimitiveRoot_adjoin_eq_top p n)
theorem

rationalCyclotomicPrincipalPrimeLocalizedPrimitiveRoot_isPrimitiveRoot

L1495 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicPrincipalPrimeLocalizedPrimitiveRoot_isPrimitiveRoot
Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · statement
theorem
    rationalCyclotomicPrincipalPrimeLocalizedPrimitiveRoot_isPrimitiveRoot
    (p : Nat.Primes) (n : ℕ) :
    IsPrimitiveRoot
      (rationalCyclotomicLocalizedPrimitiveRoot
        (rationalCyclotomicPrincipalPrimeModulus p n)
        (RayClass.rationalPrime p))
      (p.1 ^ (n + 1))
theorem

rationalCyclotomicPrincipalPrimeLocalizedPrimitiveRoot_adjoin_eq_top

L1514 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicPrincipalPrimeLocalizedPrimitiveRoot_adjoin_eq_top
Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · statement
theorem
    rationalCyclotomicPrincipalPrimeLocalizedPrimitiveRoot_adjoin_eq_top
    (p : Nat.Primes) (n : ℕ) :
    Algebra.adjoin ℚ_[p.1]
        (Set.singleton (rationalCyclotomicLocalizedPrimitiveRoot
          (rationalCyclotomicPrincipalPrimeModulus p n)
          (rationalCyclotomicArtinPlace p)) :
          Set (rationalCyclotomicArtinLocalizedField
            (rationalCyclotomicPrincipalPrimeModulus p n) p)) =
      ⊤
theorem

rationalCyclotomicPrincipalPrimeLocalizedLevel_isCyclotomicExtension

L1529 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicPrincipalPrimeLocalizedLevel_isCyclotomicExtension
Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · statement
theorem
    rationalCyclotomicPrincipalPrimeLocalizedLevel_isCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField
        (rationalCyclotomicPrincipalPrimeModulus p n) p)
def

rationalCyclotomicLocalizedCompletionPadicAlgEquiv

L1547 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicLocalizedCompletionPadicAlgEquiv
Documentation

The chosen localized global cyclotomic level, transported over the completion equivalence, is the standard multiplicative Lubin--Tate level.

Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · statement
def rationalCyclotomicLocalizedCompletionPadicAlgEquiv
    (p : Nat.Primes) (n : ℕ) :
    rationalCyclotomicArtinLocalizedField
        (rationalCyclotomicPrincipalPrimeModulus p n) p ≃ₐ[ℚ_[p.1]]
      RationalCyclotomicPrincipalPrimePadicLevel p n
theorem

rationalCyclotomicPrincipalPrime_galEquivZMod_eq_of_action

L1568 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicPrincipalPrime_galEquivZMod_eq_of_action
Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · statement
theorem
    rationalCyclotomicPrincipalPrime_galEquivZMod_eq_of_action
    (p : Nat.Primes) (n : ℕ)
    (sigma : Gal(rationalCyclotomicPrincipalPrimeLevel
      (rationalCyclotomicPrincipalPrimeModulus p n)/ℚ))
    (a : (ZMod (p.1 ^ (n + 1)))ˣ)
    (haction :
      sigma (rationalCyclotomicLevelPrimitiveRoot
          (rationalCyclotomicPrincipalPrimeModulus p n)) =
        rationalCyclotomicLevelPrimitiveRoot
            (rationalCyclotomicPrincipalPrimeModulus p n) ^ a.val.val) :
    IsCyclotomicExtension.Rat.galEquivZMod
        (p.1 ^ (n + 1))
        (rationalCyclotomicPrincipalPrimeLevel
          (rationalCyclotomicPrincipalPrimeModulus p n))
        (hK :=
          KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
            (rationalCyclotomicPrincipalPrimeModulus p n)) sigma =
      a
theorem

chosenFinitePlaceArtinMonoidHom_apply_factor_of_extension_eq

L1634 ↗
GlobalClassFieldTheory.Reciprocity.chosenFinitePlaceArtinMonoidHom_apply_factor_of_extension_eq
Documentation

The chosen finite-place Artin map factors through any extension identified with the chosen one.

Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · statement
theorem chosenFinitePlaceArtinMonoidHom_apply_factor_of_extension_eq
    {K L : Type}
    [Field K] [NumberField K]
    [Field L] [Algebra K L]
    [FiniteDimensional K L] [IsAbelianGalois K L]
    (v : HeightOneSpectrum (𝓞 K))
    (w : AbsoluteValueExtension
      (HeightOneSpectrum.adicAbv K v) L)
    (hw : chosenFinitePlaceExtension (L := L) v = w)
    (x : (v.adicCompletion K)ˣ) :
    chosenFinitePlaceArtinMonoidHom
        (K := K) (L := L) v x =
      finitePlaceLocalToGlobalMonoidHom
        (K := K) (L := L) v w
        (finitePlaceLocalArtinMonoidHom
          (K := K) (L := L) v w x)
theorem

chosenFinitePlaceArtinMonoidHom_apply_factor_of_extension_eq_at

L1658 ↗
GlobalClassFieldTheory.Reciprocity.chosenFinitePlaceArtinMonoidHom_apply_factor_of_extension_eq_at
Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · statement
theorem
    chosenFinitePlaceArtinMonoidHom_apply_factor_of_extension_eq_at
    {K L : Type}
    [Field K] [NumberField K]
    [Field L] [Algebra K L]
    [FiniteDimensional K L] [IsAbelianGalois K L]
    (v : HeightOneSpectrum (𝓞 K))
    (w : AbsoluteValueExtension
      (HeightOneSpectrum.adicAbv K v) L)
    (hw : chosenFinitePlaceExtension (L := L) v = w)
    (x : (v.adicCompletion K)ˣ) (z : L) :
    chosenFinitePlaceArtinMonoidHom
        (K := K) (L := L) v x z =
      finitePlaceLocalToGlobalMonoidHom
        (K := K) (L := L) v w
        (finitePlaceLocalArtinMonoidHom
          (K := K) (L := L) v w x) z
theorem

finitePlaceLocalToGlobalMonoidHom_apply_pow_of_localized_action

L1680 ↗
GlobalClassFieldTheory.Reciprocity.finitePlaceLocalToGlobalMonoidHom_apply_pow_of_localized_action
Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · full declaration
theorem finitePlaceLocalToGlobalMonoidHom_apply_pow_of_localized_action
    {K L : Type}
    [Field K] [NumberField K]
    [Field L] [Algebra K L]
    [FiniteDimensional K L] [IsAbelianGalois K L]
    (v : HeightOneSpectrum (𝓞 K))
    (w : AbsoluteValueExtension
      (HeightOneSpectrum.adicAbv K v) L)
    (sigma :
      let vK := HeightOneSpectrum.adicAbv K v
      let E := LocalizedCompletion vK w
      letI : Algebra vK.Completion E :=
        finitePlaceLocalArtinLocalizedAlgebra v w
      Gal(E / vK.Completion))
    (z : L)
    (zLocal :
      let vK := HeightOneSpectrum.adicAbv K v
      LocalizedCompletion vK w)
    (e : ℕ)
    (hLocalization :
      let vK := HeightOneSpectrum.adicAbv K v
      let E := LocalizedCompletion vK w
      letI : Algebra vK.Completion E :=
        finitePlaceLocalArtinLocalizedAlgebra v w
      let eLoc : L →+* E :=
        AbsoluteValue.toAlgebraicLocalization vK w.1 w.2
      eLoc z = zLocal)
    (hlocal : sigma zLocal = zLocal ^ e) :
    finitePlaceLocalToGlobalMonoidHom
        (K := K) (L := L) v w sigma z = z ^ e := by
  let vK := HeightOneSpectrum.adicAbv K v
  let hvK : vK.IsNontrivial := RayClass.adicAbv_isNontrivial v
  let E := LocalizedCompletion vK w
  let : Algebra vK.Completion E :=
    finitePlaceLocalArtinLocalizedAlgebra v w
  let eD : absoluteValueDecompositionGroup K w.1 ≃* Gal(E / vK.Completion) :=
    decompositionGroupEquivAlgebraicLocalizationAut vK hvK w
  let eLoc : L →+* E :=
    AbsoluteValue.toAlgebraicLocalization vK w.1 w.2
  let delta : absoluteValueDecompositionGroup K w.1 := eD.symm sigma
  have hDecomposition : eD delta = sigma := eD.apply_symm_apply sigma
  apply eLoc.injective
  calc
    eLoc (finitePlaceLocalToGlobalMonoidHom
        (K := K) (L := L) v w sigma z) =
        eD delta (eLoc z) :=
      (localizationRamificationGroups_decompositionGroupEquiv_toLocalization
        vK hvK w delta z).symm
    _ = sigma (eLoc z) :=
      congrArg (fun tau : Gal(E / vK.Completion) => tau (eLoc z))
        hDecomposition
    _ = sigma zLocal :=
      congrArg (fun y : E => sigma y) hLocalization
    _ = zLocal ^ e := hlocal
    _ = (eLoc z) ^ e :=
      congrArg (fun y : E => y ^ e) hLocalization.symm
    _ = eLoc (z ^ e) := (map_pow eLoc z e).symm
theorem

map_primitiveRoot_eq_pow_of_eq_pow

L1738 ↗
GlobalClassFieldTheory.Reciprocity.map_primitiveRoot_eq_pow_of_eq_pow
Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · statement
theorem map_primitiveRoot_eq_pow_of_eq_pow
    {M : Type} [CommRing M] [IsDomain M]
    (f : M →* M) (zeta rho : M) (order exponent : ℕ)
    [NeZero order]
    (hzeta : IsPrimitiveRoot zeta order)
    (hrho : IsPrimitiveRoot rho order)
    (hf : f zeta = zeta ^ exponent) :
    f rho = rho ^ exponent
theorem

finitePlaceLocalArtinMonoidHom_apply_semilinear

L1758 ↗
GlobalClassFieldTheory.Reciprocity.finitePlaceLocalArtinMonoidHom_apply_semilinear
Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · statement
theorem finitePlaceLocalArtinMonoidHom_apply_semilinear
    {K L K' L' : Type}
    [Field K] [NumberField K]
    [Field L] [Algebra K L]
    [hKLfinite : FiniteDimensional K L] [IsAbelianGalois K L]
    [Field K'] [ValuativeRel K'] [TopologicalSpace K']
    [IsNonarchimedeanLocalField K']
    [Field L'] [Algebra K' L']
    [FiniteDimensional K' L'] [IsAbelianGalois K' L']
    (v : HeightOneSpectrum (𝓞 K))
    (w : AbsoluteValueExtension
      (HeightOneSpectrum.adicAbv K v) L)
    (eK : (HeightOneSpectrum.adicAbv K v).Completion ≃+* K')
    (eL : LocalizedCompletion
      (HeightOneSpectrum.adicAbv K v) w ≃+* L')
    (hcomm : ∀ y : (HeightOneSpectrum.adicAbv K v).Completion,
      eL (@algebraMap
        (HeightOneSpectrum.adicAbv K v).Completion
        (LocalizedCompletion (HeightOneSpectrum.adicAbv K v) w)
        _ _ (finitePlaceLocalArtinLocalizedAlgebra v w) y) =
        algebraMap K' L' (eK y))
    (hExt : SemilinearValuationCompatible
      (HeightOneSpectrum.adicAbv K v).Completion K' eK)
    (x : (v.adicCompletion K)ˣ)
    (z : LocalizedCompletion (HeightOneSpectrum.adicAbv K v) w) :
    eL (finitePlaceLocalArtinMonoidHom
        (K := K) (L := L) v w x z) =
      LocalClassFieldTheory.abelianLocalArtinMonoidHom K' L'
        (Units.map eK.toMonoidHom
          (finitePlaceLocalArtinInput v x)) (eL z)
theorem

finitePlaceArtinMonoidHomOfExtension_eq_one_of_semilinear

L1840 ↗
GlobalClassFieldTheory.Reciprocity.finitePlaceArtinMonoidHomOfExtension_eq_one_of_semilinear
Documentation

If a semilinearly identified target local Artin value is trivial, then the corresponding global finite-place Artin value is trivial. This generic bridge keeps concrete completion and localization instance towers out of downstream proof terms.

Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · statement
theorem finitePlaceArtinMonoidHomOfExtension_eq_one_of_semilinear
    {K L K' L' : Type}
    [Field K] [NumberField K]
    [Field L] [Algebra K L]
    [hKLfinite : FiniteDimensional K L] [IsAbelianGalois K L]
    [Field K'] [ValuativeRel K'] [TopologicalSpace K']
    [IsNonarchimedeanLocalField K']
    [Field L'] [Algebra K' L']
    [FiniteDimensional K' L'] [IsAbelianGalois K' L']
    (v : HeightOneSpectrum (𝓞 K))
    (w : AbsoluteValueExtension
      (HeightOneSpectrum.adicAbv K v) L)
    (eK : (HeightOneSpectrum.adicAbv K v).Completion ≃+* K')
    (eL : LocalizedCompletion
      (HeightOneSpectrum.adicAbv K v) w ≃+* L')
    (hcomm : ∀ y : (HeightOneSpectrum.adicAbv K v).Completion,
      eL (@algebraMap
        (HeightOneSpectrum.adicAbv K v).Completion
        (LocalizedCompletion (HeightOneSpectrum.adicAbv K v) w)
        _ _ (finitePlaceLocalArtinLocalizedAlgebra v w) y) =
        algebraMap K' L' (eK y))
    (hExt : SemilinearValuationCompatible
      (HeightOneSpectrum.adicAbv K v).Completion K' eK)
    (x : (v.adicCompletion K)ˣ)
    (htrivial :
      LocalClassFieldTheory.abelianLocalArtinMonoidHom K' L'
          (Units.map eK.toMonoidHom
            (finitePlaceLocalArtinInput v x)) = 1) :
    finitePlaceArtinMonoidHomOfExtension
        (K := K) (L := L) v w x = 1
def

rationalCyclotomicPrincipalPrimeLocalizedRoot

L1902 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicPrincipalPrimeLocalizedRoot
Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · statement
def rationalCyclotomicPrincipalPrimeLocalizedRoot
    (p : Nat.Primes) (n : ℕ) :
    rationalCyclotomicArtinLocalizedField
      (rationalCyclotomicPrincipalPrimeModulus p n) p
def

rationalCyclotomicPrincipalPrimeResidueUnit

L1910 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicPrincipalPrimeResidueUnit
Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · statement
def rationalCyclotomicPrincipalPrimeResidueUnit
    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
    (ZMod (p.1 ^ (n + 1)))ˣ
theorem

rationalCyclotomicPrincipalPrime_localizedBase_commutes

L1920 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicPrincipalPrime_localizedBase_commutes
Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · statement
theorem rationalCyclotomicPrincipalPrime_localizedBase_commutes
    (p : Nat.Primes) (n : ℕ)
    (y : (rationalCyclotomicArtinBaseAbv p).Completion) :
    rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
        (algebraMap (rationalCyclotomicArtinBaseAbv p).Completion
          (rationalCyclotomicArtinLocalizedField
            (rationalCyclotomicPrincipalPrimeModulus p n) p) y) =
      algebraMap ℚ_[p.1]
        (RationalCyclotomicPrincipalPrimePadicLevel p n)
        (rationalFinitePlaceCompletionRingEquivPadic p y)
theorem

rationalCyclotomicPrincipalPrime_finitePlaceArtinOfExtension_eq_one_of_padic

L1959 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicPrincipalPrime_finitePlaceArtinOfExtension_eq_one_of_padic
Documentation

Specialized ramified-prime bridge from the standard p-adic Artin value to the canonical global finite-place Artin value. The localized completion and all of its dependent instances remain private to this provider.

Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · statement
theorem
    rationalCyclotomicPrincipalPrime_finitePlaceArtinOfExtension_eq_one_of_padic
    (p : Nat.Primes) (n : ℕ)
    (x : ((RayClass.rationalPrime p).adicCompletion ℚ)ˣ)
    (htrivial :
      abelianLocalArtinMonoidHom ℚ_[p.1]
          (RationalCyclotomicPrincipalPrimePadicLevel p n)
          (Units.map
            (rationalFinitePlaceCompletionRingEquivPadic p).toMonoidHom
            (finitePlaceLocalArtinInput (RayClass.rationalPrime p) x)) = 1) :
    finitePlaceArtinMonoidHomOfExtension
        (K := ℚ)
        (L := KummerTheory.rationalCyclotomicLevel
          (rationalCyclotomicPrincipalPrimeModulus p n))
        (RayClass.rationalPrime p)
        (rationalCyclotomicChosenFinitePlaceExtension
          (rationalCyclotomicPrincipalPrimeModulus p n)
          (RayClass.rationalPrime p)) x = 1
theorem

padicMultiplicativePrimitiveRoot_rationalPrimeUnitParameterGaloisAction

L1994 ↗
GlobalClassFieldTheory.Reciprocity.padicMultiplicativePrimitiveRoot_rationalPrimeUnitParameterGaloisAction
Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · statement
theorem
    padicMultiplicativePrimitiveRoot_rationalPrimeUnitParameterGaloisAction
    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
    standardLubinTateUnitParameterEquivGal
        (padicLocalField p.1)
        (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
        (standardLubinTateUnitParameterClass
          (padicLocalField p.1) n
          (rationalPrimeUnitValuationSubringUnit x p))
        (padicMultiplicativePrimitiveRoot p.1 n) =
      padicMultiplicativePrimitiveRoot p.1 n ^
        (PadicInt.toZModPow (p := p.1) (n + 1)
          (padicIntUnitOfRat p
            (rationalPrimeUnit x p : ℚ)
            (rationalPrimeUnit x p).ne_zero
            (padicValRat_rationalPrimeUnit x p) : ℤ_[p.1])).val
def

rationalCyclotomicPrincipalPrimePadicTargetArtin

L2039 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicPrincipalPrimePadicTargetArtin
Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · statement
def rationalCyclotomicPrincipalPrimePadicTargetArtin
    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
    Gal(RationalCyclotomicPrincipalPrimePadicLevel p n / ℚ_[p.1])
def

rationalCyclotomicPrincipalPrimePadicUnitParameterArtin

L2060 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicPrincipalPrimePadicUnitParameterArtin
Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · statement
def
    rationalCyclotomicPrincipalPrimePadicUnitParameterArtin
    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
    Gal(RationalCyclotomicPrincipalPrimePadicLevel p n / ℚ_[p.1])
theorem

rationalCyclotomicPrincipalPrime_padicTargetArtin_eq_unitParameter

L2071 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicPrincipalPrime_padicTargetArtin_eq_unitParameter
Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · statement
theorem
    rationalCyclotomicPrincipalPrime_padicTargetArtin_eq_unitParameter
    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
    rationalCyclotomicPrincipalPrimePadicTargetArtin p n x =
      rationalCyclotomicPrincipalPrimePadicUnitParameterArtin p n x
theorem

rationalCyclotomicPrincipalPrime_padicArtin_action_eq_unitParameter

L2105 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicPrincipalPrime_padicArtin_action_eq_unitParameter
Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · statement
theorem
    rationalCyclotomicPrincipalPrime_padicArtin_action_eq_unitParameter
    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
    (@LocalClassFieldTheory.abelianLocalArtinMonoidHom
      ℚ_[p.1] (RationalCyclotomicPrincipalPrimePadicLevel p n)
      (inferInstance : Field ℚ_[p.1])
      (inferInstance : Field
        (RationalCyclotomicPrincipalPrimePadicLevel p n))
      (inferInstance : Algebra ℚ_[p.1]
        (RationalCyclotomicPrincipalPrimePadicLevel p n))
      (inferInstance : ValuativeRel ℚ_[p.1])
      (inferInstance : TopologicalSpace ℚ_[p.1])
      (inferInstance : IsNonarchimedeanLocalField ℚ_[p.1])
      (rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional p n)
      (standardLubinTateLevelField_isAbelianGalois
        (padicLocalField p.1)
        (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n)
      (Units.map
        (rationalFinitePlaceCompletionRingEquivPadic p).toMonoidHom
        (finitePlaceLocalArtinInput
          (K := ℚ) (RayClass.rationalPrime p)
          (IdeleGroup.finiteComponent
            (RayClass.rationalPrime p)
            (IdeleGroup.principalIdele ℚ x)))))
        ((rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n).toRingEquiv
          (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)) =
      rationalCyclotomicPrincipalPrimePadicUnitParameterArtin p n x
        (rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
          (rationalCyclotomicPrincipalPrimeLocalizedRoot p n))
theorem

rationalCyclotomicPrincipalPrime_padicUnitParameterArtin_action

L2187 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicPrincipalPrime_padicUnitParameterArtin_action
Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · statement
theorem
    rationalCyclotomicPrincipalPrime_padicUnitParameterArtin_action
    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
    rationalCyclotomicPrincipalPrimePadicUnitParameterArtin p n x
        (rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
          (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)) =
      (rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
        (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)) ^
        (rationalCyclotomicPrincipalPrimeResidueUnit p n x).val.val
theorem

rationalCyclotomicPrincipalPrime_localArtin_action

L2226 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicPrincipalPrime_localArtin_action
Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · statement
theorem rationalCyclotomicPrincipalPrime_localArtin_action
    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
    finitePlaceLocalArtinMonoidHom
        (K := ℚ)
        (L := KummerTheory.rationalCyclotomicLevel
          (rationalCyclotomicPrincipalPrimeModulus p n))
        (RayClass.rationalPrime p)
        (rationalCyclotomicChosenFinitePlaceExtension
          (rationalCyclotomicPrincipalPrimeModulus p n)
          (RayClass.rationalPrime p))
        (IdeleGroup.finiteComponent
          (RayClass.rationalPrime p)
          (IdeleGroup.principalIdele ℚ x))
        (rationalCyclotomicPrincipalPrimeLocalizedRoot p n) =
      (rationalCyclotomicPrincipalPrimeLocalizedRoot p n) ^
        (rationalCyclotomicPrincipalPrimeResidueUnit p n x).val.val
def

rationalCyclotomicPrincipalPrimeChosenArtin

L2280 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicPrincipalPrimeChosenArtin
Documentation

The finite-place Artin symbol at the ramified prime, in its canonical local-to-global factored form. Keeping this specialization opaque prevents its dependent local/global instance tower from being unfolded downstream.

Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · statement
def rationalCyclotomicPrincipalPrimeChosenArtin
    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
    KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n) ≃ₐ[ℚ]
      KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)
theorem

galEquivZMod_chosenFinitePlaceArtinMonoidHom_principal_at_prime

L2308 ↗
GlobalClassFieldTheory.Reciprocity.galEquivZMod_chosenFinitePlaceArtinMonoidHom_principal_at_prime
Documentation

At the ramified prime, the cyclotomic character of the chosen finite-place Artin symbol is the direct reduction of the rational p-adic unit.

Source context 61 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
section RationalCyclotomicPrincipalPrime
noncomputable local instance
    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
    (p : Nat.Primes) (n : ℕ) :
    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
  change
    IsCyclotomicExtension
      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
      (KummerTheory.rationalCyclotomicLevel
        (rationalCyclotomicPrincipalPrimeModulus p n))
  exact
    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
      (rationalCyclotomicPrincipalPrimeModulus p n)
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicAlgebra
    (m : ℕ+) (p : Nat.Primes) :
    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
noncomputable local instance
    rationalCyclotomicArtinLocalizedPadicScalarTower
    (m : ℕ+) (p : Nat.Primes) :
    IsScalarTower ℚ ℚ_[p.1]
      (rationalCyclotomicArtinLocalizedField m p) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
attribute [local instance]
  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
noncomputable local instance
    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
    (p : Nat.Primes) (n : ℕ) :
    IsAbelianGalois ℚ_[p.1]
      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
  standardLubinTateLevelField_isAbelianGalois
    (padicLocalField p.1)
    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
@[reducible]
noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
    (p : Nat.Primes) :
    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
Lean source · statement
theorem galEquivZMod_chosenFinitePlaceArtinMonoidHom_principal_at_prime
    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
    IsCyclotomicExtension.Rat.galEquivZMod
        (p.1 ^ (n + 1))
        (KummerTheory.rationalCyclotomicLevel
          (rationalCyclotomicPrincipalPrimeModulus p n))
        (hK :=
          KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
            (rationalCyclotomicPrincipalPrimeModulus p n))
        (rationalCyclotomicPrincipalPrimeChosenArtin p n x) =
      Units.map
        (PadicInt.toZModPow (p := p.1) (n + 1)).toMonoidHom
        (padicIntUnitOfRat p
          (rationalPrimeUnit x p : ℚ)
          (rationalPrimeUnit x p).ne_zero
          (padicValRat_rationalPrimeUnit x p))
theorem

rationalCyclotomicArtinUnramified

L2334 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinUnramified
Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
theorem rationalCyclotomicArtinUnramified
    (m : ℕ+) (q : Nat.Primes)
    (hq : ¬ q.1 ∣ (m : ℕ)) :
    IsNonarchimedeanLocalField.IsUnramifiedValuedExtension
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q)
def

rationalCyclotomicArtinLocalFrobeniusOf

L2344 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalFrobeniusOf
Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
def rationalCyclotomicArtinLocalFrobeniusOf
    (m : ℕ+) (q : Nat.Primes)
    (hUnramified :
      IsNonarchimedeanLocalField.IsUnramifiedValuedExtension
        (rationalCyclotomicArtinBaseAbv q).Completion
        (rationalCyclotomicArtinLocalizedField m q)) :
    rationalCyclotomicArtinLocalizedField m q ≃ₐ[
      (rationalCyclotomicArtinBaseAbv q).Completion]
      rationalCyclotomicArtinLocalizedField m q
def

rationalCyclotomicArtinLocalFrobenius

L2358 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalFrobenius
Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
def rationalCyclotomicArtinLocalFrobenius
    (m : ℕ+) (q : Nat.Primes)
    (hq : ¬ q.1 ∣ (m : ℕ)) :
    rationalCyclotomicArtinLocalizedField m q ≃ₐ[
      (rationalCyclotomicArtinBaseAbv q).Completion]
      rationalCyclotomicArtinLocalizedField m q
def

rationalCyclotomicArtinDecompositionEquiv

L2367 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinDecompositionEquiv
Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
def rationalCyclotomicArtinDecompositionEquiv
    (m : ℕ+) (q : Nat.Primes) :
    absoluteValueDecompositionGroup ℚ
        (rationalCyclotomicArtinExtension m q).1 ≃*
      (rationalCyclotomicArtinLocalizedField m q ≃ₐ[
        (rationalCyclotomicArtinBaseAbv q).Completion]
        rationalCyclotomicArtinLocalizedField m q)
def

rationalCyclotomicArtinLocalToGlobalMonoidHom

L2380 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalToGlobalMonoidHom
Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
def rationalCyclotomicArtinLocalToGlobalMonoidHom
    (m : ℕ+) (q : Nat.Primes) :
    (rationalCyclotomicArtinLocalizedField m q ≃ₐ[
      (rationalCyclotomicArtinBaseAbv q).Completion]
      rationalCyclotomicArtinLocalizedField m q) →*
        (rationalCyclotomicArtinLevel m ≃ₐ[ℚ]
          rationalCyclotomicArtinLevel m)
abbrev

rationalCyclotomicArtinLocalArtin

L2392 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalArtin
Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
abbrev rationalCyclotomicArtinLocalArtin
    (m : ℕ+) (q : Nat.Primes)
    (x : ((RayClass.rationalPrime q).adicCompletion ℚ)ˣ) :
    rationalCyclotomicArtinLocalizedField m q ≃ₐ[
      (rationalCyclotomicArtinBaseAbv q).Completion]
      rationalCyclotomicArtinLocalizedField m q
def

rationalCyclotomicArtinGlobalFrobeniusOf

L2403 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinGlobalFrobeniusOf
Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
def rationalCyclotomicArtinGlobalFrobeniusOf
    (m : ℕ+) (q : Nat.Primes)
    (hUnramified :
      IsNonarchimedeanLocalField.IsUnramifiedValuedExtension
        (rationalCyclotomicArtinBaseAbv q).Completion
        (rationalCyclotomicArtinLocalizedField m q)) :
    rationalCyclotomicArtinLevel m ≃ₐ[ℚ]
      rationalCyclotomicArtinLevel m
def

rationalCyclotomicChosenArithmeticFrobenius

L2418 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicChosenArithmeticFrobenius
Documentation

The global decomposition-group lift of arithmetic Frobenius at the chosen place above q. Keeping the local construction opaque prevents its many completion instances from leaking into later theorem statements.

Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
def rationalCyclotomicChosenArithmeticFrobenius
    (m : ℕ+) (q : Nat.Primes)
    (hq : ¬ q.1 ∣ (m : ℕ)) :
      KummerTheory.rationalCyclotomicLevel m ≃ₐ[ℚ]
      KummerTheory.rationalCyclotomicLevel m
abbrev

rationalCyclotomicArtinLocalInput

L2427 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalInput
Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
abbrev rationalCyclotomicArtinLocalInput
    (q : Nat.Primes)
    (x : ((RayClass.rationalPrime q).adicCompletion ℚ)ˣ) :
    (rationalCyclotomicArtinBaseAbv q).Completionˣ
def

rationalCyclotomicArtinLocalExponent

L2437 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalExponent
Documentation

The normalized valuation of the canonical completion input used by the rational finite-place Artin map. This named endpoint keeps the completion instances out of downstream theorem statements.

Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
def rationalCyclotomicArtinLocalExponent
    (q : Nat.Primes)
    (x : ((RayClass.rationalPrime q).adicCompletion ℚ)ˣ) : ℤ
def

rationalCyclotomicChosenFinitePlaceArtinValue

L2446 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicChosenFinitePlaceArtinValue
Documentation

The chosen finite-place Artin value in a rational cyclotomic level, with the completion and Galois instance arguments frozen at the provider boundary.

Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
def rationalCyclotomicChosenFinitePlaceArtinValue
    (m : ℕ+) (q : Nat.Primes)
    (x : ((RayClass.rationalPrime q).adicCompletion ℚ)ˣ) :
    Gal(KummerTheory.rationalCyclotomicLevel m / ℚ)
theorem

rationalCyclotomicArtinLocalArtin_eq

L2455 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalArtin_eq
Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
theorem rationalCyclotomicArtinLocalArtin_eq
    (m : ℕ+) (q : Nat.Primes)
    (x : ((RayClass.rationalPrime q).adicCompletion ℚ)ˣ) :
    rationalCyclotomicArtinLocalArtin m q x =
      @LocalClassFieldTheory.abelianLocalArtinMonoidHom
        (rationalCyclotomicArtinBaseAbv q).Completion
        (rationalCyclotomicArtinLocalizedField m q)
        (inferInstance : Field
          (rationalCyclotomicArtinBaseAbv q).Completion)
        (inferInstance : Field
          (rationalCyclotomicArtinLocalizedField m q))
        (finitePlaceLocalArtinLocalizedAlgebra
          (K := ℚ) (L := rationalCyclotomicArtinLevel m)
          (rationalCyclotomicArtinPlace q)
          (rationalCyclotomicArtinExtension m q))
        (finitePlaceLocalArtinCompletionValuativeRel
          (K := ℚ) (rationalCyclotomicArtinPlace q))
        (inferInstance : TopologicalSpace
          (rationalCyclotomicArtinBaseAbv q).Completion)
        (finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
          (K := ℚ) (rationalCyclotomicArtinPlace q))
        (finitePlaceLocalArtinFiniteDimensional
          (K := ℚ) (L := rationalCyclotomicArtinLevel m)
          (rationalCyclotomicArtinPlace q)
          (rationalCyclotomicArtinExtension m q))
        (finitePlaceLocalArtinIsAbelianGalois
          (K := ℚ) (L := rationalCyclotomicArtinLevel m)
          (rationalCyclotomicArtinPlace q)
          (rationalCyclotomicArtinExtension m q)
          (inferInstance : FiniteDimensional ℚ
            (rationalCyclotomicArtinLevel m)))
        (finitePlaceLocalArtinInput
          (K := ℚ) (rationalCyclotomicArtinPlace q) x)
theorem

rationalCyclotomicChosenArithmeticFrobenius_eq_lift

L2490 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicChosenArithmeticFrobenius_eq_lift
Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
theorem rationalCyclotomicChosenArithmeticFrobenius_eq_lift
    (m : ℕ+) (q : Nat.Primes)
    (hq : ¬ q.1 ∣ (m : ℕ)) :
    rationalCyclotomicChosenArithmeticFrobenius m q hq =
      (absoluteValueDecompositionGroup ℚ
        (rationalCyclotomicArtinExtension m q).1).subtype
        ((rationalCyclotomicArtinDecompositionEquiv m q).symm
          (rationalCyclotomicArtinLocalFrobenius m q hq))
theorem

rationalCyclotomicFinitePlaceMappedLocalArtin_eq_frobenius_zpow_of

L2500 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicFinitePlaceMappedLocalArtin_eq_frobenius_zpow_of
Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
theorem
    rationalCyclotomicFinitePlaceMappedLocalArtin_eq_frobenius_zpow_of
    (m : ℕ+) (q : Nat.Primes)
    (hAbelian :
      IsAbelianGalois
        (rationalCyclotomicArtinBaseAbv q).Completion
        (rationalCyclotomicArtinLocalizedField m q))
    (hUnramified :
      IsNonarchimedeanLocalField.IsUnramifiedValuedExtension
        (rationalCyclotomicArtinBaseAbv q).Completion
        (rationalCyclotomicArtinLocalizedField m q))
    (x : ((RayClass.rationalPrime q).adicCompletion ℚ)ˣ) :
    rationalCyclotomicArtinLocalToGlobalMonoidHom m q
        (LocalClassFieldTheory.abelianLocalArtinMonoidHom
          (rationalCyclotomicArtinBaseAbv q).Completion
          (rationalCyclotomicArtinLocalizedField m q)
          (rationalCyclotomicArtinLocalInput q x)) =
      (rationalCyclotomicArtinGlobalFrobeniusOf
        m q hUnramified) ^
        rationalCyclotomicArtinLocalExponent q x
theorem

rationalCyclotomicFinitePlaceMappedLocalArtin_eq_frobenius_zpow

L2540 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicFinitePlaceMappedLocalArtin_eq_frobenius_zpow
Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
theorem
    rationalCyclotomicFinitePlaceMappedLocalArtin_eq_frobenius_zpow
    (m : ℕ+) (q : Nat.Primes)
    (hq : ¬ q.1 ∣ (m : ℕ))
    (x : ((RayClass.rationalPrime q).adicCompletion ℚ)ˣ) :
    rationalCyclotomicArtinLocalToGlobalMonoidHom m q
        (rationalCyclotomicArtinLocalArtin m q x) =
      (rationalCyclotomicChosenArithmeticFrobenius m q hq) ^
        rationalCyclotomicArtinLocalExponent q x
theorem

chosenFinitePlaceArtin_eq_chosenArithmeticFrobenius_zpow

L2577 ↗
GlobalClassFieldTheory.Reciprocity.chosenFinitePlaceArtin_eq_chosenArithmeticFrobenius_zpow
Documentation

The chosen local Artin symbol is the chosen global Frobenius lift raised to the normalized local valuation.

Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
theorem
    chosenFinitePlaceArtin_eq_chosenArithmeticFrobenius_zpow
    (m : ℕ+) (q : Nat.Primes)
    (hq : ¬ q.1 ∣ (m : ℕ))
    (x : ((RayClass.rationalPrime q).adicCompletion ℚ)ˣ) :
    chosenFinitePlaceArtinMonoidHom
        (K := ℚ) (L := rationalCyclotomicArtinLevel m)
        (rationalCyclotomicArtinPlace q) x =
      (rationalCyclotomicChosenArithmeticFrobenius m q hq) ^
        rationalCyclotomicArtinLocalExponent q x
theorem

rationalCyclotomicArtinResidueFieldCard

L2629 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinResidueFieldCard
Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
theorem rationalCyclotomicArtinResidueFieldCard
    (q : Nat.Primes) :
    Nat.card 𝓀[(rationalCyclotomicArtinBaseAbv q).Completion] = q.1
theorem

rationalCyclotomicArtinLocalFrobenius_apply_root

L2637 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalFrobenius_apply_root
Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
theorem rationalCyclotomicArtinLocalFrobenius_apply_root
    (m : ℕ+) (q : Nat.Primes)
    (hq : ¬ q.1 ∣ (m : ℕ)) :
    rationalCyclotomicArtinLocalFrobenius m q hq
        (rationalCyclotomicLocalizedPrimitiveRoot
          m (rationalCyclotomicArtinPlace q)) =
      (rationalCyclotomicLocalizedPrimitiveRoot
        m (rationalCyclotomicArtinPlace q)) ^ q.1
theorem

rationalCyclotomicArtinFrobeniusLift_localization

L2668 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinFrobeniusLift_localization
Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
theorem rationalCyclotomicArtinFrobeniusLift_localization
    (m : ℕ+) (q : Nat.Primes)
    (hq : ¬ q.1 ∣ (m : ℕ)) :
    AbsoluteValue.toAlgebraicLocalization
        (rationalCyclotomicArtinBaseAbv q)
        (rationalCyclotomicArtinExtension m q).1
        (rationalCyclotomicArtinExtension m q).2
        (((absoluteValueDecompositionGroup ℚ
          (rationalCyclotomicArtinExtension m q).1).subtype
          ((rationalCyclotomicArtinDecompositionEquiv m q).symm
            (rationalCyclotomicArtinLocalFrobenius m q hq)))
          (rationalCyclotomicLevelPrimitiveRoot m)) =
      rationalCyclotomicArtinLocalFrobenius m q hq
        (AbsoluteValue.toAlgebraicLocalization
          (rationalCyclotomicArtinBaseAbv q)
          (rationalCyclotomicArtinExtension m q).1
          (rationalCyclotomicArtinExtension m q).2
          (rationalCyclotomicLevelPrimitiveRoot m))
theorem

rationalCyclotomicChosenArithmeticFrobenius_localization

L2730 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicChosenArithmeticFrobenius_localization
Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
theorem rationalCyclotomicChosenArithmeticFrobenius_localization
    (m : ℕ+) (q : Nat.Primes)
    (hq : ¬ q.1 ∣ (m : ℕ)) :
    AbsoluteValue.toAlgebraicLocalization
        (rationalCyclotomicArtinBaseAbv q)
        (rationalCyclotomicArtinExtension m q).1
        (rationalCyclotomicArtinExtension m q).2
        (rationalCyclotomicChosenArithmeticFrobenius m q hq
          (rationalCyclotomicLevelPrimitiveRoot m)) =
      rationalCyclotomicArtinLocalFrobenius m q hq
        (AbsoluteValue.toAlgebraicLocalization
          (rationalCyclotomicArtinBaseAbv q)
          (rationalCyclotomicArtinExtension m q).1
          (rationalCyclotomicArtinExtension m q).2
          (rationalCyclotomicLevelPrimitiveRoot m))
theorem

rationalCyclotomicArtinPrimitiveRoot_localization

L2778 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinPrimitiveRoot_localization
Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
theorem rationalCyclotomicArtinPrimitiveRoot_localization
    (m : ℕ+) (q : Nat.Primes) :
    AbsoluteValue.toAlgebraicLocalization
        (rationalCyclotomicArtinBaseAbv q)
        (rationalCyclotomicArtinExtension m q).1
        (rationalCyclotomicArtinExtension m q).2
        (rationalCyclotomicLevelPrimitiveRoot m) =
      rationalCyclotomicLocalizedPrimitiveRoot
        m (rationalCyclotomicArtinPlace q)
theorem

rationalCyclotomicArtinLocalizedRoot_pow

L2789 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicArtinLocalizedRoot_pow
Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
theorem rationalCyclotomicArtinLocalizedRoot_pow
    (m : ℕ+) (q : Nat.Primes) :
    (rationalCyclotomicLocalizedPrimitiveRoot
        m (rationalCyclotomicArtinPlace q)) ^ q.1 =
      rationalCyclotomicGlobalToLocalizedAlgHom
        m (rationalCyclotomicArtinPlace q)
        (rationalCyclotomicLevelPrimitiveRoot m ^ q.1)
theorem

rationalCyclotomicChosenArithmeticFrobenius_apply_root

L2813 ↗
GlobalClassFieldTheory.Reciprocity.rationalCyclotomicChosenArithmeticFrobenius_apply_root
Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
theorem rationalCyclotomicChosenArithmeticFrobenius_apply_root
    (m : ℕ+) (q : Nat.Primes)
    (hq : ¬ q.1 ∣ (m : ℕ)) :
    rationalCyclotomicChosenArithmeticFrobenius m q hq
        (rationalCyclotomicLevelPrimitiveRoot m) =
      rationalCyclotomicLevelPrimitiveRoot m ^ q.1
theorem

galEquivZMod_chosenArithmeticFrobenius

L2885 ↗
GlobalClassFieldTheory.Reciprocity.galEquivZMod_chosenArithmeticFrobenius
Documentation

The cyclotomic character sends the chosen arithmetic Frobenius lift to the residue prime.

Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
theorem galEquivZMod_chosenArithmeticFrobenius
    (m : ℕ+) (q : Nat.Primes)
    (hq : ¬ q.1 ∣ (m : ℕ)) :
    IsCyclotomicExtension.Rat.galEquivZMod
        (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
        (rationalCyclotomicChosenArithmeticFrobenius m q hq) =
      ZMod.unitOfCoprime q.1
        (q.2.coprime_iff_not_dvd.mpr hq)
theorem

galEquivZMod_chosenFinitePlaceArtinMonoidHom_of_not_dvd

L2901 ↗
GlobalClassFieldTheory.Reciprocity.galEquivZMod_chosenFinitePlaceArtinMonoidHom_of_not_dvd
Documentation

At a rational prime not dividing the level, the cyclotomic character of the chosen finite-place Artin symbol is the residue prime raised to the normalized local valuation.

Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · full declaration
theorem galEquivZMod_chosenFinitePlaceArtinMonoidHom_of_not_dvd
    (m : ℕ+) (q : Nat.Primes)
    (hq : ¬ q.1 ∣ (m : ℕ))
    (x : ((RayClass.rationalPrime q).adicCompletion ℚ)ˣ) :
    let v : HeightOneSpectrum (𝓞 ℚ) :=
      RayClass.rationalPrime q
    let L := KummerTheory.rationalCyclotomicLevel m
    let vQ := HeightOneSpectrum.adicAbv ℚ v
    let localInput :=
      (finitePlaceCompletionUnitsContinuousMulEquiv v).symm x
    let localExponent :=
      LocalFieldTheory.IsNonarchimedeanLocalField.valuationMap
        vQ.Completion (Additive.ofMul localInput)
    IsCyclotomicExtension.Rat.galEquivZMod
        (m : ℕ) L
        (chosenFinitePlaceArtinMonoidHom
          (K := ℚ)
          (L := L) v x) =
      (ZMod.unitOfCoprime q.1
        (q.2.coprime_iff_not_dvd.mpr hq)) ^ localExponent := by
  dsimp only
  rw [chosenFinitePlaceArtin_eq_chosenArithmeticFrobenius_zpow
      m q hq x, map_zpow,
    galEquivZMod_chosenArithmeticFrobenius m q hq]
  rfl
theorem

galEquivZMod_chosenFinitePlaceArtinMonoidHom_eq_one_of_not_dvd_of_localExponent_eq_zero

L2929 ↗
GlobalClassFieldTheory.Reciprocity.galEquivZMod_chosenFinitePlaceArtinMonoidHom_eq_one_of_not_dvd_of_localExponent_eq_zero
Documentation

Away from the cyclotomic level, a finite-place input of normalized valuation zero has trivial cyclotomic character.

Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
theorem
    galEquivZMod_chosenFinitePlaceArtinMonoidHom_eq_one_of_not_dvd_of_localExponent_eq_zero
    (m : ℕ+) (q : Nat.Primes)
    (hq : ¬ q.1 ∣ (m : ℕ))
    (x : ((RayClass.rationalPrime q).adicCompletion ℚ)ˣ)
    (hzero : rationalCyclotomicArtinLocalExponent q x = 0) :
    IsCyclotomicExtension.Rat.galEquivZMod
        (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
        (chosenFinitePlaceArtinMonoidHom
          (K := ℚ)
          (L := KummerTheory.rationalCyclotomicLevel m)
          (RayClass.rationalPrime q) x) =
      1
theorem

chosenFinitePlaceArtinMonoidHom_eq_one_of_not_dvd_of_localExponent_eq_zero

L2952 ↗
GlobalClassFieldTheory.Reciprocity.chosenFinitePlaceArtinMonoidHom_eq_one_of_not_dvd_of_localExponent_eq_zero
Documentation

Away from the cyclotomic level, valuation zero makes the chosen finite-place Artin symbol itself trivial. Returning the Galois element, rather than an equality between cyclotomic characters with frozen instance arguments, lets downstream restriction arguments apply their own canonical character without a dependent instance transport.

Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
theorem
    chosenFinitePlaceArtinMonoidHom_eq_one_of_not_dvd_of_localExponent_eq_zero
    (m : ℕ+) (q : Nat.Primes)
    (hq : ¬ q.1 ∣ (m : ℕ))
    (x : ((RayClass.rationalPrime q).adicCompletion ℚ)ˣ)
    (hzero : rationalCyclotomicArtinLocalExponent q x = 0) :
    chosenFinitePlaceArtinMonoidHom
        (K := ℚ)
        (L := KummerTheory.rationalCyclotomicLevel m)
        (RayClass.rationalPrime q) x =
      1
theorem

galEquivZMod_chosenFinitePlaceArtinMonoidHom_principal_of_not_dvd

L2973 ↗
GlobalClassFieldTheory.Reciprocity.galEquivZMod_chosenFinitePlaceArtinMonoidHom_principal_of_not_dvd
Documentation

For a rational principal idele, the unramified finite-place cyclotomic Artin symbol at q is q raised to the negative usual q-adic exponent.

Source context 54 commands

Surrounding namespaces, section variables, and local commands at this declaration.

open scoped NNReal NumberField ValuativeRel
open NumberField IsDedekindDomain
noncomputable section
namespace GlobalClassFieldTheory
namespace Reciprocity

-- Specializing the generic finite-place comparison to `ℚ` must retain its
-- `Algebra.id` owner rather than selecting the competing rational-field
-- instance introduced after specialization.
@[reducible] noncomputable local instance
    rationalFinitePlaceCompletionRatAlgebra
    (v : HeightOneSpectrum (𝓞 ℚ)) :
    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
  letI : Algebra ℚ ℚ := Algebra.id ℚ
  let hWith : Algebra ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instAlgebra _
  let hUniform : UniformContinuousConstSMul ℚ
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
    WithAbs.instUniformContinuousConstSMulReal _
  exact
    @UniformSpace.Completion.algebra
      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
      ℚ _ hWith hUniform
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open LocalClassFieldTheory
open LocalFieldTheory
open LocalFieldTheory.DiscreteValuationField
open LocalFieldTheory.DiscreteValuationField.Examples.Qp
open LubinTate
local instance (q : Nat.Primes) : Fact q.1.Prime :=
  ⟨q.2⟩
local instance (m : ℕ+) : NeZero (m : ℕ) :=
  ⟨m.ne_zero⟩
noncomputable local instance
    rationalCyclotomicLevelFiniteDimensional
    (m : ℕ+) :
    FiniteDimensional ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
  IsCyclotomicExtension.finiteDimensional
    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
noncomputable local instance
    rationalCyclotomicLevelIsAbelianGalois
    (m : ℕ+) :
    IsAbelianGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) := by
  have : IsGalois ℚ
      (KummerTheory.rationalCyclotomicLevel m) :=
    inferInstance
  let e :=
    IsCyclotomicExtension.Rat.galEquivZMod
      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
  exact
    { is_comm.comm σ τ := by
        apply e.injective
        simp only [map_mul]
        exact mul_comm _ _ }
@[reducible]
noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
    (q : Nat.Primes) :
    NontriviallyNormedField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  absoluteValueExtension_completionNontriviallyNormedField
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (RayClass.adicAbv_isNontrivial
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
    (q : Nat.Primes) :
    LocallyCompactSpace
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  AbsoluteValue.Completion.locallyCompactSpace
    (finitePlaceCompletionBaseMap_isometry
      (RayClass.rationalPrime q))
noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
    (q : Nat.Primes) :
    IsUltrametricDist
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceArtinCompletionIsUltrametricDist
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValued
    (q : Nat.Primes) :
    Valued
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
  finitePlaceArtinCompletionValued
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (RayClass.rationalPrime q))
@[reducible]
noncomputable local instance rationalFinitePlaceBaseValuativeRel
    (q : Nat.Primes) :
    ValuativeRel
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionValuativeRel
    (K := ℚ) (RayClass.rationalPrime q)
noncomputable local instance
    rationalFinitePlaceBaseValuationIsNontrivial
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).IsNontrivial :=
  (inferInstance :
    (NormedField.valuation
      (K :=
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
noncomputable local instance rationalFinitePlaceBaseValuationCompatible
    (q : Nat.Primes) :
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalFinitePlaceBaseValuativeRelIsNontrivial
    (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (Valued.v :
      Valuation
        (HeightOneSpectrum.adicAbv ℚ
          (RayClass.rationalPrime q)).Completion
        ℝ≥0)).2 inferInstance
noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
    (q : Nat.Primes) :
    IsValuativeTopology
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  isValuativeTopology_of_valued_ofValuation
    (HeightOneSpectrum.adicAbv ℚ
      (RayClass.rationalPrime q)).Completion ℝ≥0
noncomputable local instance
    rationalFinitePlaceBaseIsNonarchimedeanLocalField
    (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (HeightOneSpectrum.adicAbv ℚ
        (RayClass.rationalPrime q)).Completion :=
  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
    (K := ℚ) (RayClass.rationalPrime q)
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.extensionCompletionAlgebra
    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
@[reducible]
noncomputable local instance rationalCyclotomicArtinExtensionSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinCompletionAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  AbsoluteValue.completionAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q).1
    (rationalCyclotomicArtinExtension m q).2
@[reducible]
noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinLocalizedAlgebra
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedGlobalSMul
    (m : ℕ+) (q : Nat.Primes) :
    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
noncomputable local instance
    rationalCyclotomicArtinLocalizedScalarTower
    (m : ℕ+) (q : Nat.Primes) :
    IsScalarTower ℚ
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) := by
  constructor
  intro r x y
  simp only [Algebra.smul_def, map_mul, eq_ratCast,
    map_ratCast, mul_assoc]
noncomputable local instance
    rationalCyclotomicArtinLocalizedFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinFiniteDimensional
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsAbelianGalois
    (m : ℕ+) (q : Nat.Primes) :
    IsAbelianGalois
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  finitePlaceLocalArtinIsAbelianGalois
    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
    (rationalCyclotomicArtinPlace q)
    (rationalCyclotomicArtinExtension m q)
    (inferInstance :
      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsSeparable
    (m : ℕ+) (q : Nat.Primes) :
    Algebra.IsSeparable
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsCyclotomic
    (m : ℕ+) (q : Nat.Primes) :
    IsCyclotomicExtension {(m : ℕ)}
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinLocalizedField m q) :=
  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
    m (rationalCyclotomicArtinPlace q)
noncomputable local instance
    rationalCyclotomicArtinExtensionFiniteDimensional
    (m : ℕ+) (q : Nat.Primes) :
    FiniteDimensional
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  completionModuleFinite
    (rationalCyclotomicArtinBaseAbv q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (rationalCyclotomicArtinExtension m q)
noncomputable local instance
    rationalCyclotomicArtinExtensionContinuousSMul
    (m : ℕ+) (q : Nat.Primes) :
    ContinuousSMul
      (rationalCyclotomicArtinBaseAbv q).Completion
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  continuousSMul_of_algebraMap _ _
    (AbsoluteValue.completionMap_isometry
      (rationalCyclotomicArtinBaseAbv q)
      (rationalCyclotomicArtinExtension m q).1
      (rationalCyclotomicArtinExtension m q).2).continuous
noncomputable local instance
    rationalCyclotomicArtinExtensionLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinExtension m q).1.Completion :=
  LocallyCompactSpace.of_finiteDimensional_of_complete
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinExtension m q).1.Completion
noncomputable local instance
    rationalCyclotomicArtinLocalizedLocallyCompact
    (m : ℕ+) (q : Nat.Primes) :
    LocallyCompactSpace
      (rationalCyclotomicArtinLocalizedField m q) :=
  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
    inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsUltrametricDist
    (m : ℕ+) (q : Nat.Primes) :
    IsUltrametricDist
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsUltrametricDist
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance rationalCyclotomicArtinLocalizedValued
    (m : ℕ+) (q : Nat.Primes) :
    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
  localizedCompletionFinitePlaceValued
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
@[reducible]
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRel
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionFinitePlaceValuativeRel
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationCompatible
    (m : ℕ+) (q : Nat.Primes) :
    (Valued.v : Valuation
      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
  Valuation.Compatible.ofValuation _
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationHasExtension
    (m : ℕ+) (q : Nat.Primes) :
    Valuation.HasExtension
      (ValuativeRel.valuation
        (rationalCyclotomicArtinBaseAbv q).Completion)
      (ValuativeRel.valuation
        (rationalCyclotomicArtinLocalizedField m q)) :=
  localizedCompletionValuationHasExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuationIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
  Valuation.IsNontrivial.of_hasExtension
    (ValuativeRel.valuation
      (rationalCyclotomicArtinBaseAbv q).Completion)
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
    (m : ℕ+) (q : Nat.Primes) :
    ValuativeRel.IsNontrivial
      (rationalCyclotomicArtinLocalizedField m q) :=
  (ValuativeRel.isNontrivial_iff_isNontrivial
    (ValuativeRel.valuation
      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsValuativeTopology
    (m : ℕ+) (q : Nat.Primes) :
    IsValuativeTopology
      (rationalCyclotomicArtinLocalizedField m q) :=
  isValuativeTopology_of_valued_ofValuation
    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
    (m : ℕ+) (q : Nat.Primes) :
    IsNonarchimedeanLocalField
      (rationalCyclotomicArtinLocalizedField m q) :=
  { toIsValuativeTopology := inferInstance
    toLocallyCompactSpace := inferInstance
    toIsNontrivial := inferInstance }
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerAlgebra
    (m : ℕ+) (q : Nat.Primes) :
    Algebra
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  Algebra.ofSubsemiring
    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
noncomputable local instance
    rationalCyclotomicArtinLocalizedIsIntegralClosure
    (m : ℕ+) (q : Nat.Primes) :
    IsIntegralClosure
      𝒪[rationalCyclotomicArtinLocalizedField m q]
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      (rationalCyclotomicArtinLocalizedField m q) :=
  localizedCompletionIsIntegralClosureWithExtension
    (rationalCyclotomicArtinBaseAbv q)
    (rationalCyclotomicArtinExtension m q)
    (RayClass.adicAbv_isNontrivial
      (rationalCyclotomicArtinPlace q))
    (HeightOneSpectrum.isNonarchimedean_adicAbv
      ℚ (rationalCyclotomicArtinPlace q))
noncomputable local instance
    rationalCyclotomicArtinLocalizedIntegerModuleFinite
    (m : ℕ+) (q : Nat.Primes) :
    Module.Finite
      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
  integerRing_moduleFinite_of_isIntegralClosure
    (rationalCyclotomicArtinBaseAbv q).Completion
    (rationalCyclotomicArtinLocalizedField m q)
Lean source · statement
theorem
    galEquivZMod_chosenFinitePlaceArtinMonoidHom_principal_of_not_dvd
    (m : ℕ+) (q : Nat.Primes)
    (hq : ¬ q.1 ∣ (m : ℕ)) (x : ℚˣ) :
    IsCyclotomicExtension.Rat.galEquivZMod
        (m : ℕ)
        (KummerTheory.rationalCyclotomicLevel m)
        (chosenFinitePlaceArtinMonoidHom
          (K := ℚ)
          (L := KummerTheory.rationalCyclotomicLevel m)
          (RayClass.rationalPrime q)
          (IdeleGroup.finiteComponent
            (RayClass.rationalPrime q)
            (IdeleGroup.principalIdele ℚ x))) =
      (ZMod.unitOfCoprime q.1
        (q.2.coprime_iff_not_dvd.mpr hq)) ^
          (-padicValRat q.1 (x : ℚ))

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