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 from the monograph [1] 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.