Jacobian varieties
A Jacobian gives a curve's degree-zero divisor theory the structure of an algebraic group. The library constructs this object over the curve's original field, even when the curve has no rational point, and proves that its dimension equals the genus. Choosing a rational point then gives an Abel–Jacobi morphism through which every pointed map to an abelian variety factors uniquely.
Over algebraically closed fields of characteristic zero, the later results count torsion points and first étale cohomology, describe the contribution of punctures, and compare the abelianized fundamental group with the full Tate module. The Tate comparison assumes a representing Picard object and compatibility with divisor classes; the Weil-pairing result proves nondegeneracy for supplied divisorial pairing data.
Selected theorems and definitions
- definition
A Jacobian without a rational base point
A smooth proper geometrically irreducible curve has a fixed descended Jacobian over its original field.
- theorem
The group scheme and its dimension
The same descended Jacobian is smooth, proper, geometrically irreducible, and has dimension equal to the curve's genus.
- definition
The pointed Abel–Jacobi morphism
A chosen rational point determines a morphism from the curve to its fixed Jacobian and maps to the identity.
- theorem
Unique factorization through Abel–Jacobi
Every pointed morphism from the curve to an abelian variety factors uniquely through the fixed Jacobian.
- theorem
A pointed genus-one curve is its Jacobian
In characteristic zero, the constructed Abel–Jacobi map for a genus-one curve is an isomorphism.
- theorem
Counting torsion points
An abelian variety of dimension d over an algebraically closed field of characteristic zero has n^(2d) points killed by n.
- theorem
Finite étale H¹ of a proper curve
The actual first étale cohomology group with roots-of-unity coefficients has the expected genus-dependent cardinality.
- theorem
The contribution of punctures to H¹
For a nonempty finite boundary, the first étale cohomology count gains r−1 boundary directions.
- theorem
Abel–Jacobi and the full Tate module
Under explicit Picard representation and class-compatibility hypotheses, Abel–Jacobi identifies the abelianized fundamental group with the full Tate module.
- theorem
Nondegeneracy of the divisorial Weil pairing
Every nonzero torsion divisor class is detected by a character obtained from supplied divisorial Weil-pairing data.
References used in these guides
- J. S. Milne. Jacobian Varieties. Springer, New York, in Arithmetic Geometry (Storrs, Conn., 1984), Chapter VII, 167–212, 1986. Author's revised version dated June 12, 2021; original numbering retained.
- J. S. Milne. Abelian Varieties. Springer, New York, in Arithmetic Geometry (Storrs, Conn., 1984), Chapter V, 103–150, 1986. Author's revised version dated January 2, 2022; original numbering retained.
- Alexander Grothendieck et al.. Revêtements étales et groupe fondamental (SGA 1). Springer, Lecture Notes in Mathematics 224, 1971. Link to the updated edition with remarks by Michel Raynaud.
- Alexander Grothendieck. Technique de descente et théorèmes d'existence en géométrie algébrique. V. Les schémas de Picard : théorèmes d'existence. Séminaire Bourbaki, Exposé 232, 143–161, 1962.