Potential Sha for abelian varieties

TL;DR

This paper proves that the p-torsion part of the Tate-Shafarevich group for any principally polarized abelian variety over a number field becomes unbounded when considering extensions of degree proportional to p, with the constant depending only on the variety's dimension.

Contribution

It establishes the unboundedness of p-torsion in the Tate-Shafarevich group over certain extensions, a new result in the arithmetic of abelian varieties.

Findings

p-torsion in Tate-Shafarevich group is unbounded over extensions of degree O(p)

The result depends only on the dimension of the abelian variety

Provides new insights into the arithmetic of abelian varieties over number fields

Abstract

We show that the p-torsion in the Tate-Shafarevich group of any principally polarized abelian variety over a number field is unbounded as one ranges over extensions of degree O(p), the implied constant depending only on the dimension of the abelian variety.