Module

Place_sum_ramification_evalAt_eq.lean

Nyll.AlgebraicCurves.Places.Place_sum_ramification_evalAt_eq

31 declarations indexed from sourceView source ↗

Module documentation

Place sum ramification eval At eq

Imports11

Imported by1

Nyll.AlgebraicCurves.Places.All

def

fib

L101 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.fib

Documentation

The set of places at which the selected function is regular and takes the prescribed value.

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)

Lean source · statement

def fib (b : ℂ) : Set (Place ℂ F)

def

poles

L104 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.poles

Documentation

The set of poles of the selected function.

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)

Lean source · statement

def poles : Set (Place ℂ F)

def

eFin

L107 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.eFin

Documentation

The nonnegative multiplicity of the selected function minus the prescribed finite value.

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)

Lean source · statement

def eFin (w : Place ℂ F) (b : ℂ) : ℕ

def

eInf

L110 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.eInf

Documentation

The nonnegative order of the inverse function at a place.

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)

Lean source · statement

def eInf (w : Place ℂ F) : ℕ

abbrev

deg

L114 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.deg

Documentation

The degree of the function-field extension over the field generated by the selected function.

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)

Lean source · statement

abbrev deg : ℕ

theorem

mem_fib_iff

L116 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.mem_fib_iff

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)

Lean source · statement

theorem mem_fib_iff {w : Place ℂ F} {b : ℂ} :
    w ∈ fib x b ↔ x ∈ w.toValuationSubring ∧ Place.evalAt w x = b

theorem

mem_poles_iff

L119 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.mem_poles_iff

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)

Lean source · statement

theorem mem_poles_iff {w : Place ℂ F} : w ∈ poles x ↔ x ∉ w.toValuationSubring

theorem

evalAt_sub_algebraMap

L121 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.evalAt_sub_algebraMap

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)

Lean source · statement

theorem evalAt_sub_algebraMap (v : Place ℂ F) (hv : v.IsRational) {f : F}
    (hf : f ∈ v.toValuationSubring) (c : ℂ) :
    v.evalAt (f - algebraMap ℂ F c) = v.evalAt f - c

theorem

evalAt_eq_zero_of_ord_pos

L133 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.evalAt_eq_zero_of_ord_pos

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)

Lean source · statement

theorem evalAt_eq_zero_of_ord_pos (v : Place ℂ F) {g : F} (hg : g ≠ 0)
    (h : 0 < v.ord g) : v.evalAt g = 0

theorem

ord_eq_zero_of_evalAt_ne_zero

L149 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.ord_eq_zero_of_evalAt_ne_zero

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)

Lean source · statement

theorem ord_eq_zero_of_evalAt_ne_zero (v : Place ℂ F) {g : F} (hg : g ≠ 0)
    (hmem : g ∈ v.toValuationSubring) (hne : v.evalAt g ≠ 0) : v.ord g = 0

theorem

ord_pos_of_evalAt_eq_zero

L156 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.ord_pos_of_evalAt_eq_zero

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)

Lean source · statement

theorem ord_pos_of_evalAt_eq_zero (v : Place ℂ F) (hv : v.IsRational) {g : F} (hg : g ≠ 0)
    (hmem : g ∈ v.toValuationSubring) (hz : v.evalAt g = 0) : 0 < v.ord g

theorem

ord_multiset_prod

L163 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.ord_multiset_prod

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)

Lean source · statement

theorem ord_multiset_prod (v : Place ℂ F) (s : Multiset F) (hs : ∀ y ∈ s, y ≠ 0) :
    v.ord s.prod = (s.map v.ord).sum

theorem

ord_algebraMap_eq_zero

L174 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.ord_algebraMap_eq_zero

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)

Lean source · statement

theorem ord_algebraMap_eq_zero (v : Place ℂ F) {c : ℂ} (hc : c ≠ 0) :
    v.ord (algebraMap ℂ F c) = 0

theorem

sum_map_ite_count

L189 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.sum_map_ite_count

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)

Lean source · full declaration

theorem sum_map_ite_count (t : ℂ) (e : ℤ) (s : Multiset ℂ) :
    (s.map (fun r => if r = t then e else 0)).sum = (s.count t : ℤ) * e := by
  classical
  induction s using Multiset.induction_on with
  | empty => simp
  | cons a s ih =>
    rw [Multiset.map_cons, Multiset.sum_cons, ih, Multiset.count_cons]
    by_cases h : a = t
    · rw [ite_eq_left h, ite_eq_left h.symm]
      push_cast
      ring
    · rw [ite_eq_right h, ite_eq_right (fun hh => h hh.symm)]
      push_cast
      ring

theorem

ord_aeval

L204 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.ord_aeval

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)

Lean source · statement

theorem ord_aeval (v : Place ℂ F) (hv : v.IsRational) {g : F} (htg : Transcendental ℂ g)
    (hmem : g ∈ v.toValuationSubring) {p : ℂ[X]} (hp : p ≠ 0) :
    v.ord (Polynomial.aeval g p)
      = v.ord (g - algebraMap ℂ F (v.evalAt g)) * p.rootMultiplicity (v.evalAt g)

theorem

isRational_of_finite

L249 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.isRational_of_finite

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)

Lean source · statement

theorem isRational_of_finite {L : Type*} [Field L] [Algebra ℂ L] (v : Place ℂ L)
    (hfin : Module.Finite ℂ v.ResidueField) : v.IsRational

theorem

transcendental_inv

L255 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.transcendental_inv

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)

Lean source · statement

theorem transcendental_inv {g : F} (hg : Transcendental ℂ g) : Transcendental ℂ g⁻¹

theorem

adjoin_le_adjoin_inv

L258 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.adjoin_le_adjoin_inv

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)

Lean source · statement

theorem adjoin_le_adjoin_inv (g : F) :
    IntermediateField.adjoin ℂ ({g} : Set F) ≤ IntermediateField.adjoin ℂ ({g⁻¹} : Set F)

theorem

inv_mem_of_mem_poles

L264 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.inv_mem_of_mem_poles

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)

Lean source · statement

theorem inv_mem_of_mem_poles {w : Place ℂ F} (hw : w ∈ poles x) : x⁻¹ ∈ w.toValuationSubring

theorem

ne_zero_of_mem_poles

L267 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.ne_zero_of_mem_poles

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)

Lean source · statement

theorem ne_zero_of_mem_poles {w : Place ℂ F} (hw : w ∈ poles x) : x ≠ 0

theorem

ord_neg_of_mem_poles

L273 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.ord_neg_of_mem_poles

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)

Lean source · statement

theorem ord_neg_of_mem_poles {w : Place ℂ F} (hw : w ∈ poles x) : w.ord x < 0

theorem

ord_inv_pos_of_mem_poles

L278 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.ord_inv_pos_of_mem_poles

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)

Lean source · statement

theorem ord_inv_pos_of_mem_poles {w : Place ℂ F} (hw : w ∈ poles x) : 0 < w.ord x⁻¹

theorem

evalAt_inv_of_mem_poles

L283 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.evalAt_inv_of_mem_poles

Source context 11 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)

Lean source · statement

theorem evalAt_inv_of_mem_poles {w : Place ℂ F} (hw : w ∈ poles x) : Place.evalAt w x⁻¹ = 0

theorem

place_isRational

L289 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.place_isRational

Source context 12 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)
variable [IsCurveOver ℂ F]

Lean source · statement

theorem place_isRational (w : Place ℂ F) : w.IsRational

theorem

exists_ord_coe_eq

L292 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.exists_ord_coe_eq

Source context 12 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)
variable [IsCurveOver ℂ F]

Lean source · statement

theorem exists_ord_coe_eq {g : F} (htg : Transcendental ℂ g)
    (hle : IntermediateField.adjoin ℂ ({x} : Set F) ≤ IntermediateField.adjoin ℂ ({g} : Set F))
    (c : ℂ) (y : ↥(IntermediateField.adjoin ℂ ({x} : Set F))) (hy : y ≠ 0) :
    ∃ k : ℤ, ∀ w : Place ℂ F, g ∈ w.toValuationSubring → w.evalAt g = c →
      w.ord (y : F) = w.ord (g - algebraMap ℂ F c) * k

theorem

ramificationIndex_eq_ord

L322 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.ramificationIndex_eq_ord

Source context 12 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)
variable [IsCurveOver ℂ F]

Lean source · statement

theorem ramificationIndex_eq_ord
    [FiniteDimensional (IntermediateField.adjoin ℂ ({x} : Set F)) F]
    {g : F} (htg : Transcendental ℂ g)
    (hle : IntermediateField.adjoin ℂ ({x} : Set F) ≤ IntermediateField.adjoin ℂ ({g} : Set F))
    (hgE : g ∈ IntermediateField.adjoin ℂ ({x} : Set F)) (c : ℂ) (w : Place ℂ F)
    (hgm : g ∈ w.toValuationSubring) (hgc : w.evalAt g = c) :
    (w.ramificationIndex (↥(IntermediateField.adjoin ℂ ({x} : Set F))) : ℤ)
      = w.ord (g - algebraMap ℂ F c)

theorem

eFin_eq_ramificationIndex

L358 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.eFin_eq_ramificationIndex

Source context 12 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)
variable [IsCurveOver ℂ F]

Lean source · statement

theorem eFin_eq_ramificationIndex (htr : Transcendental ℂ x)
    [FiniteDimensional (IntermediateField.adjoin ℂ ({x} : Set F)) F] {w : Place ℂ F} {b : ℂ}
    (hw : w ∈ fib x b) :
    (eFin x w b : ℤ) = (w.ramificationIndex (↥(IntermediateField.adjoin ℂ ({x} : Set F))) : ℤ)

theorem

eInf_eq_ramificationIndex

L371 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.eInf_eq_ramificationIndex

Source context 12 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)
variable [IsCurveOver ℂ F]

Lean source · statement

theorem eInf_eq_ramificationIndex (htr : Transcendental ℂ x)
    [FiniteDimensional (IntermediateField.adjoin ℂ ({x} : Set F)) F] {w : Place ℂ F}
    (hw : w ∈ poles x) :
    (eInf x w : ℤ) = (w.ramificationIndex (↥(IntermediateField.adjoin ℂ ({x} : Set F))) : ℤ)

theorem

rh_count

L389 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RamificationByValue.rh_count

Source context 14 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
noncomputable section
namespace RamificationByValue
open AlgebraicCurve Polynomial
variable {F : Type*} [Field F] [Algebra ℂ F]
variable (x : F)
variable [IsCurveOver ℂ F]
section RiemannHurwitz
variable [HasCanonicalDivisor (K := ℂ) (F := F)]

Lean source · statement

theorem rh_count (htr : Transcendental ℂ x)
    [FiniteDimensional (IntermediateField.adjoin ℂ ({x} : Set F)) F] (B : Finset ℂ)
    (hB : ∀ b w, w ∈ fib x b → eFin x w b ≠ 1 → b ∈ B)
    (t : ℂ → Finset (Place ℂ F)) (ht : ∀ b ∈ B, ∀ w, w ∈ t b ↔ w ∈ fib x b)
    (p : Finset (Place ℂ F)) (hp : ∀ w, w ∈ p ↔ w ∈ poles x) :
    (∑ b ∈ B, ∑ w ∈ t b, ((eFin x w b : ℤ) - 1)) + ∑ w ∈ p, ((eInf x w : ℤ) - 1)
      = 2 * (Module.finrank ℂ ↥(regularDifferentials ℂ F) : ℤ) - 2 + 2 * (deg x : ℤ)

theorem

sum_ramification_evalAt_eq

L476 ↗

AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq.RiemannHurwitzByValue.sum_ramification_evalAt_eq

Source context 9 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
section MainStatement
open AlgebraicCurve IntermediateField
namespace RiemannHurwitzByValue

Lean source · statement

theorem sum_ramification_evalAt_eq {F : Type*} [Field F] [Algebra ℂ F] [IsCurveOver ℂ F]
    [HasCanonicalDivisor (K := ℂ) (F := F)]
    {x : F} (htr : Transcendental ℂ x) [FiniteDimensional (↥(adjoin ℂ ({x} : Set F))) F]
    (B : Finset ℂ)
    (hB : ∀ (b : ℂ) (w : Place ℂ F), x ∈ w.toValuationSubring → Place.evalAt w x = b →
      (w.ord (x - algebraMap ℂ F b)).toNat ≠ 1 → b ∈ B)
    (t : ℂ → Finset (Place ℂ F))
    (ht : ∀ b ∈ B, ∀ w : Place ℂ F, w ∈ t b ↔ x ∈ w.toValuationSubring ∧ Place.evalAt w x = b)
    (p : Finset (Place ℂ F)) (hp : ∀ w : Place ℂ F, w ∈ p ↔ x ∉ w.toValuationSubring) :
    (∑ b ∈ B, ∑ w ∈ t b, (((w.ord (x - algebraMap ℂ F b)).toNat : ℤ) - 1)) +
        ∑ w ∈ p, (((w.ord x⁻¹).toNat : ℤ) - 1) =
      2 * (Module.finrank ℂ ↥(regularDifferentials ℂ F) : ℤ) - 2 +
        2 * (Module.finrank (↥(adjoin ℂ ({x} : Set F))) F : ℤ)

theorem

sum_ramification_evalAt_eq

L499 ↗

AlgebraicCurve.Place.sum_ramification_evalAt_eq

Source context 7 commands

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

section ProofImplementation
namespace AlgebraicCurves.Places.Proofs.Place_sum_ramification_evalAt_eq
attribute [-instance]
  instIsRankOneDiscreteWithZeroMultiplicativeIntAdicValuation
  instIsTrivialOnWithZeroMultiplicativeIntAdicValuation
  AlgebraicCurve.instHasLocalResidue_of_hasCanonicalLocalResidueK
  AlgebraicCurve.instHasCanonicalLocalResidueK_of_hasCanonicalLocalResidueKStar
  kw_ffgc_finiteDimensional_adicCompletion
  kw_ffgc_isScalarTower_integersIntegersCompletion
  ModularCurve.KwF4gRRTate.instAlgebraKAdicCompletionIntegers
  kw_ffgc_continuousSMul_adicCompletionComap
  kw_ffgc_isScalarTower_integersCompletionCompletion
  instLiesOverSubtypeAdicCompletionMemValuationSubringAdicCompletionIntegersCompletionIdealAsIdeal
  AlgebraicCurve.instHasCanonicalLocalResidueK
  instAlgebra_restrictResidueField
  instIsScalarTower_restrictResidueField
  AlgebraicCurve.instHasLocalResidue
  AlgebraicCurve.HasSeparableResidue.of_perfectField_of_isCurveOver
  AlgebraicCurve.HasSeparableResidue.of_perfectField
  instIsLocalHom_restrictSubringHom
  AlgebraicCurve.instHasCanonicalLocalResidueKStar
  ModularCurve.KwNo6Pin.isLocalRing_completion
  AlgebraicCurve.RationalFunctionField.instNontrivialInftyValueGroup
  instIsPrimeCenter
  instIsFractionRingIntegralClosureAt
  instIsTorsionFreeSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
  instIsDedekindDomainIntegralClosureAt
  instFiniteSubtypeMemValuationSubringToValuationSubringIntegralClosureAt
attribute [-simp] AlgebraicCurve.Place.congrEquiv_symm_apply
  AlgebraicCurve.RationalFunctionField.heightOneSpectrumOfIrreducible_asIdeal
  AlgebraicCurve.Place.congrRingEquiv_toValuationSubring AlgebraicCurve.Place.congrEquiv_apply
  AlgebraicCurve.Place.coe_comapSymmRingEquiv_apply
  AlgebraicCurve.RationalFunctionField.deg_placeOfPoint
attribute [-simp] AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.sizeOf_spec
  AlgebraicCurve.Place.CanonicalLocalResidueDataK.mk.injEq AlgebraicCurve.adeleSingle_coe
  AlgebraicCurve.kaehlerResidueTermKFam_apply AlgebraicCurve.Place.LocalResidueData.mk.injEq
  AlgebraicCurve.Place.LocalResidueData.mk.sizeOf_spec AlgebraicCurve.mulAdele_apply
  AlgebraicCurve.residuePairing_apply_coe AlgebraicCurve.mem_adeleBdd AlgebraicCurve.weilSmul_one
  AlgebraicCurve.diagonalHom_apply AlgebraicCurve.weilSmul_apply AlgebraicCurve.adeleSpaceMul_coe
  AlgebraicCurve.mulAdele_one AlgebraicCurve.Place.kw_ffgc_adicCompletionComapIntegers_coe
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.sizeOf_spec
  AlgebraicCurve.Place.mem_simplePoleSubmodule AlgebraicCurve.Place.coe_uniformizerSubring
  ModularCurve.Lg37.Lg37CompletionSection.mk.injEq
  AlgebraicCurve.Place.CoefficientFieldSection.mk.injEq
  AlgebraicCurve.Place.CanonicalLocalResidueDataS.mk.injEq
  ModularCurve.Lg37.Lg37CompletionSection.mk.sizeOf_spec
  AlgebraicCurve.Place.CoefficientFieldSection.mk.sizeOf_spec
  AlgebraicCurve.Place.poleSubmodule_one AlgebraicCurve.Place.mem_poleSubmodule
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeInfty
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_some
  AlgebraicCurve.RationalFunctionField.placeEquivOption_placeOfPoint
  AlgebraicCurve.RationalFunctionField.placeEquivOption_symm_none
attribute [-simp] AlgebraicCurve.RationalFunctionField.placeInfty_toValuationSubring
  AlgebraicCurve.Place.placeOfPrime_toValuationSubring AlgebraicCurve.Place.mem_fiberOver
  AlgebraicCurve.Place.fiberEquiv_symm_apply AlgebraicCurve.Place.fiberEquiv_apply
  AlgebraicCurve.Place.centerHeightOneSpectrum_asIdeal
  AlgebraicCurve.TranscendenceTower.mk.sizeOf_spec AlgebraicCurve.IntegralBasisInLSpace.mk.injEq
  AlgebraicCurve.TranscendenceTower.mk.injEq AlgebraicCurve.PoleDivisorPackage.mk.sizeOf_spec
  AlgebraicCurve.IntegralBasisInLSpace.mk.sizeOf_spec AlgebraicCurve.PoleDivisorPackage.mk.injEq
open AlgebraicCurve IntermediateField

Lean source · statement

theorem _root_.AlgebraicCurve.Place.sum_ramification_evalAt_eq {F : Type*} [Field F] [Algebra ℂ F]
  [IsCurveOver ℂ F]
    [HasCanonicalDivisor (K := ℂ) (F := F)]
    {x : F} (htr : Transcendental ℂ x) [FiniteDimensional (↥(adjoin ℂ ({x} : Set F))) F]
    (B : Finset ℂ)
    (hB : ∀ (b : ℂ) (w : Place ℂ F), x ∈ w.toValuationSubring → Place.evalAt w x = b →
      (w.ord (x - algebraMap ℂ F b)).toNat ≠ 1 → b ∈ B)
    (t : ℂ → Finset (Place ℂ F))
    (ht : ∀ b ∈ B, ∀ w : Place ℂ F, w ∈ t b ↔ x ∈ w.toValuationSubring ∧ Place.evalAt w x = b)
    (p : Finset (Place ℂ F)) (hp : ∀ w : Place ℂ F, w ∈ p ↔ x ∉ w.toValuationSubring) :
    (∑ b ∈ B, ∑ w ∈ t b, (((w.ord (x - algebraMap ℂ F b)).toNat : ℤ) - 1)) +
        ∑ w ∈ p, (((w.ord x⁻¹).toNat : ℤ) - 1) =
      2 * (Module.finrank ℂ ↥(regularDifferentials ℂ F) : ℤ) - 2 +
        2 * (Module.finrank (↥(adjoin ℂ ({x} : Set F))) F : ℤ)

Module documentation

Place sum ramification eval At eq

Declarations