Arbitrarily large $p$-torsion in Tate-Shafarevich groups

TL;DR

This paper demonstrates the existence of simple abelian varieties over the rationals with arbitrarily large p-torsion in their Tate-Shafarevich groups by constructing explicit covers that violate the Hasse principle.

Contribution

It provides explicit constructions of abelian varieties with large p-torsion in Tate-Shafarevich groups, advancing understanding of their structure and properties.

Findings

Existence of abelian varieties with arbitrarily large p-torsion

Explicit $rac{p}{rac{p}{rac{p}{rac{p}{rac{p}{rac{p}{rac{p}{rac{p}{rac{p}{rac{p}{rac{p}{rac{p}{rac{p}{rac{p}{rac{p}{rac{p}{rac{p}{rac{p}{rac{p}{rac{p/ covers of Jacobians that violate the Hasse principle

Interpretation of the proof in terms of the Cassels-Tate pairing

Abstract

We show that, for any prime $p$, there exist absolutely simple abelian varieties over $\mathbb{Q}$ with arbitrarily large $p$-torsion in their Tate-Shafarevich group. To prove this, we construct explicit $\mu_p$-covers of Jacobians of the form $y^p = x(x-1)(x-a)$ which violate the Hasse principle. In the appendix, Tom Fisher explains how to interpret our proof in terms of a Cassels-Tate pairing.