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

TL;DR

This paper determines the structure of Tate-Shafarevich groups for certain elliptic curves over global function fields, revealing their decomposition into cyclic groups influenced by Drinfeld-Heegner points.

Contribution

It provides a detailed description of the Tate-Shafarevich groups' structure over global function fields, linking their decomposition to Drinfeld-Heegner points.

Findings

Tate-Shafarevich groups are finite abelian groups.

Decomposition into cyclic groups depends on Drinfeld-Heegner points.

Explicit structure of these groups is established.

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.