Weil-Ch\^atelet divisible elements in Tate-Shafarevich groups I: The Bashmakov problem for elliptic curves over Q

TL;DR

This paper investigates the structure of divisible elements in the Tate-Shafarevich group of elliptic curves over Q, focusing on the maximal divisible subgroup of the first Galois cohomology group and its intersection with Sha.

Contribution

It provides a detailed analysis of the maximal divisible subgroup in H^1(k,A) for elliptic curves over Q, addressing the Bashmakov problem in this context.

Findings

Characterization of the divisible subgroup in H^1(k,A)

Results on the intersection with Sha(A/k) for elliptic curves over Q

Advances in understanding the structure of Tate-Shafarevich groups

Abstract

For an abelian variety A over a number field k we discuss the maximal divisibile subgroup of H^1(k,A) and its intersection with the subgroup Sha(A/k). The results are most complete for elliptic curves over Q.