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

E. Victor Flynn; Ari Shnidman

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

Number Theory 2024-10-30 v2 Algebraic Geometry

Authors: E. Victor Flynn , Ari Shnidman

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.

Keywords

abelian varieties algebraic varieties abelian groups

Cite

@article{arxiv.2209.08088,
  title  = {Arbitrarily large $p$-torsion in Tate-Shafarevich groups},
  author = {E. Victor Flynn and Ari Shnidman},
  journal= {arXiv preprint arXiv:2209.08088},
  year   = {2024}
}

Comments

Added an appendix by Tom Fisher and made a few minor changes. To appear in J. Inst. Math. Jussieu

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

Abstract

Keywords

Cite

Comments

Related papers