Abelian variety

A complex torus of dimension g is a torus of real dimension 2g that carries the structure of a complex manifold. It can always be obtained as the quotient of a g-dimensional complex vector space by a lattice of rank 2g. A complex abelian variety of dimension g is a complex torus of dimension g that is also a projective algebraic variety over the field of complex numbers. By invoking the Kodaira embedding theorem and Chow's theorem one may equivalently define a complex abelian variety of dimension g to be a complex torus of dimension g that admits a positive line bundle. Since they are complex tori, abelian varieties carry the structure of a group. A morphism of abelian varieties is a morphism of the underlying algebraic varieties that preserves the identity element for the group structure. An isogeny is a finite-to-one morphism.

When a complex torus carries the structure of an algebraic variety, this structure is necessarily unique. In the case g = 1, the notion of abelian variety is the same as that of elliptic curve, and every complex torus gives rise to such a curve; for g > 1 it has been known since Riemann that the algebraic variety condition imposes extra constraints on a complex torus.

The following criterion by Riemann decides whether or not a given complex torus is an abelian variety, i.e. whether or not it can be embedded into a projective space. Let X be a g-dimensional torus given as X = V/L where V is a complex vector space of dimension g and L is a lattice in V. Then X is an abelian variety if and only if there exists a positive definite hermitian form on V whose imaginary part takes integral values on L×L. Such a form on X is usually called a (non-degenerate) Riemann form. Choosing a basis for V and L, one can make this condition more explicit. There are several equivalent formulations of this; all of them are known as the Riemann conditions.

Every algebraic curve C of genus g ≥ 1 is associated with an abelian variety J of dimension g, by means of an analytic map of C into J. As a torus, J carries a commutative group structure, and the image of C generates J as a group. More accurately, J is covered by C^g:[1] any point in J comes from a g-tuple of points in C. The study of differential forms on C, which give rise to the abelian integrals with which the theory started, can be derived from the simpler, translation-invariant theory of differentials on J. The abelian variety J is called the Jacobian variety of C, for any non-singular curve C over the complex numbers. From the point of view of birational geometry, its function field is the fixed field of the symmetric group on g letters acting on the function field of C^g.

An abelian function is a meromorphic function on an abelian variety, which may be regarded therefore as a periodic function of n complex variables, having 2n independent periods; equivalently, it is a function in the function field of an abelian variety. For example, in the nineteenth century there was much interest in hyperelliptic integrals that may be expressed in terms of elliptic integrals. This comes down to asking that J is a product of elliptic curves, up to an isogeny.

One important structure theorem of abelian varieties is Matsusaka's theorem. It states that over an algebraically closed field every abelian variety ${\displaystyle A}$ is the quotient of the Jacobian of some curve; that is, there is some surjection of abelian varieties ${\displaystyle J\to A}$ where ${\displaystyle J}$ is a Jacobian. This theorem remains true if the ground field is infinite.[2]

To an abelian variety A over a field k, one associates a dual abelian variety A^v (over the same field), which is the solution to the following moduli problem. A family of degree 0 line bundles parametrised by a k-variety T is defined to be a line bundle L on A×T such that

1. for all t in T, the restriction of L to A×{t} is a degree 0 line bundle,
2. the restriction of L to {0}×T is a trivial line bundle (here 0 is the identity of A).

Then there is a variety A^v and a family of degree 0 line bundles P, the Poincaré bundle, parametrised by A^v such that a family L on T is associated a unique morphism f: T → A^v so that L is isomorphic to the pullback of P along the morphism 1_A×f: A×T → A×A^v. Applying this to the case when T is a point, we see that the points of A^v correspond to line bundles of degree 0 on A, so there is a natural group operation on A^v given by tensor product of line bundles, which makes it into an abelian variety.

This association is a duality in the sense that there is a natural isomorphism between the double dual A^vv and A (defined via the Poincaré bundle) and that it is contravariant functorial, i.e. it associates to all morphisms f: A → B dual morphisms f^v: B^v → A^v in a compatible way. The n-torsion of an abelian variety and the n-torsion of its dual are dual to each other when n is coprime to the characteristic of the base. In general - for all n - the n-torsion group schemes of dual abelian varieties are Cartier duals of each other. This generalises the Weil pairing for elliptic curves.

A polarisation of an abelian variety is an isogeny from an abelian variety to its dual that is symmetric with respect to double-duality for abelian varieties and for which the pullback of the Poincaré bundle along the associated graph morphism is ample (so it is analogous to a positive-definite quadratic form). Polarised abelian varieties have finite automorphism groups. A principal polarisation is a polarisation that is an isomorphism. Jacobians of curves are naturally equipped with a principal polarisation as soon as one picks an arbitrary rational base point on the curve, and the curve can be reconstructed from its polarised Jacobian when the genus is > 1. Not all principally polarised abelian varieties are Jacobians of curves; see the Schottky problem. A polarisation induces a Rosati involution on the endomorphism ring ${\displaystyle \mathrm {End} (A)\otimes \mathbb {Q} }$ of A.

Over the complex numbers, a polarised abelian variety can also be defined as an abelian variety A together with a choice of a Riemann form H. Two Riemann forms H₁ and H₂ are called equivalent if there are positive integers n and m such that nH₁=mH₂. A choice of an equivalence class of Riemann forms on A is called a polarisation of A. A morphism of polarised abelian varieties is a morphism A → B of abelian varieties such that the pullback of the Riemann form on B to A is equivalent to the given form on A.

Let ${\displaystyle A,B\in \mathbb {Z} }$ be such that ${\displaystyle x^{3}+Ax+B}$ has no repeated complex roots. Then the discriminant ${\displaystyle \Delta =-16(4A^{3}+27B^{2})}$ is nonzero. Let ${\displaystyle R=\mathbb {Z} [1/\Delta ]}$, so ${\displaystyle \operatorname {Spec} R}$ is an open subscheme of ${\displaystyle \operatorname {Spec} \mathbb {Z} }$. Then ${\displaystyle \operatorname {Proj} R[x,y,z]/(y^{2}z-x^{3}-Axz^{2}-Bz^{3})}$ is an abelian scheme over ${\displaystyle \operatorname {Spec} R}$. It can be extended to a Néron model over ${\displaystyle \operatorname {Spec} \mathbb {Z} }$, which is a smooth group scheme over ${\displaystyle \operatorname {Spec} \mathbb {Z} }$, but the Néron model is not proper and hence is not an abelian scheme over ${\displaystyle \operatorname {Spec} \mathbb {Z} }$.

V. A. Abrashkin[3] and Jean-Marc Fontaine[4] independently proved that there are no nonzero abelian varieties over Q with good reduction at all primes. Equivalently, there are no nonzero abelian schemes over Spec Z. The proof involves showing that the coordinates of pⁿ-torsion points generate number fields with very little ramification and hence of small discriminant, while, on the other hand, there are lower bounds on discriminants of number fields.[5]