Rational Points On Abelian Varieties
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.
Read more about this topic: Arithmetic Of Abelian Varieties
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