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Concept
Arithmetic of abelian varieties
Summary
In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to the studies of Pierre de Fermat on what are now recognized as elliptic curves; and has become a very substantial area of arithmetic geometry both in terms of results and conjectures. Most of these can be posed for an abelian variety A over a number field K; or more generally (for global fields or more general finitely-generated rings or fields).
There is some tension here between concepts: integer point belongs in a sense to affine geometry, while abelian variety is inherently defined in projective geometry. The basic results, such as Siegel's theorem on integral points, come from the theory of diophantine approximation.
The basic result, the Mordell–Weil theorem in Diophantine geometry, says that A(K), the group of points on A over K, is a finitely-generated abelian group. A great deal of information about its possible torsion subgroups is known, at least when A is an elliptic curve. The question of the rank is thought to be bound up with L-functions (see below).
The torsor theory here leads to the Selmer group and Tate–Shafarevich group, the latter (conjecturally finite) being difficult to study.
Height function
The theory of heights plays a prominent role in the arithmetic of abelian varieties. For instance, the canonical Néron–Tate height is a quadratic form with remarkable properties that appear in the statement of the Birch and Swinnerton-Dyer conjecture.
Reduction of an abelian variety A modulo a prime ideal of (the integers of) K — say, a prime number p — to get an abelian variety Ap over a finite field, is possible for almost all p. The 'bad' primes, for which the reduction degenerates by acquiring singular points, are known to reveal very interesting information. As often happens in number theory, the 'bad' primes play a rather active role in the theory.
Here a refined theory of (in effect) a right adjoint to reduction mod p — the Néron model — cannot always be avoided.
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