Arithmetic of elliptic curves

In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or family of those. It goes back to the studies of Fermat on what are now recognised as elliptic curves; and has become a very substantial area 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 proving that elliptic curves have finitely many integer points come out of diophantine approximation.

The basic result (Mordell–Weil theorem) 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.

There is a canonical Néron–Tate height function, which is a quadratic form; it has some remarkable properties, amongst all height functions designed to specify finite sets in A(K) of points of height (roughly, logarithmic size of co-ordinates) at most h.

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