Structure of Tate–Shafarevich groups of elliptic curves over global function fields

Abstract

The structure of the Tate–Shafarevich groups of a class of elliptic curves over global function fields is determined. These are known to be finite abelian groups and hence they are direct sums of finite cyclic groups where the orders of these cyclic components are invariants of the Tate–Shafarevich group. This decomposition of the Tate–Shafarevich groups into direct sums of finite cyclic groups depends on the behaviour of Drinfeld–Heegner points on these elliptic curves. These are points analogous to Heegner points on elliptic curves over the rational numbers.