A note on the order of the Tate--Shafarevich group modulo squares

TL;DR

This paper demonstrates that any square-free natural number can appear as the non-square-free part of the Tate--Shafarevich group of some abelian variety, confirming a conjecture by W. Stein.

Contribution

It proves that all square-free numbers can be realized as the non-square-free part of the Tate--Shafarevich group, validating W. Stein's conjecture.

Findings

Every square-free number appears as the non-square-free part of some Tate--Shafarevich group.

The result confirms a long-standing conjecture in the field.

Provides new insight into the structure of Tate--Shafarevich groups.

Abstract

We investigate the order of the Tate--Shafarevich group of abelian varieties modulo rational squares. Our main result shows that every square-free natural number appears as the non square-free part of the Tate--Shafarevich group of some abelian variety, thereby validating a conjecture of W. Stein.