Weil-Ch\^atelet divisible elements in Tate-Shafarevich groups II: On a question of Cassels

TL;DR

This paper investigates the divisibility properties of elements in the Tate-Shafarevich group of an abelian variety over a number field, with detailed results for elliptic curves over the rationals, addressing a question posed by Cassels.

Contribution

It provides new insights into the divisibility of Sha elements in H^1(k,A), especially for elliptic curves over Q, advancing understanding of Cassels' question.

Findings

Enhanced understanding of divisibility in Sha for elliptic curves over Q

Partial classification of divisible elements in Tate-Shafarevich groups

Connections established between divisibility and Cassels' question

Abstract

For an abelian variety A over a number field k we discuss the divisibility in H^1(k,A) of elements of the subgroup Sha(A/k). The results are most complete for elliptic curves over Q.