Explicit $7$-torsion in the Tate-Shafarevich groups of genus $2$ Jacobians

TL;DR

The paper presents an algorithm to identify genus 2 Jacobians with non-trivial 7-torsion in their Tate-Shafarevich groups, demonstrating this phenomenon explicitly for certain curves with real multiplication.

Contribution

It introduces a new algorithm to find twists of the Klein quartic that relate to elliptic curves with specific Galois representation properties, revealing explicit 7-torsion elements in Tate-Shafarevich groups.

Findings

Identified genus 2 Jacobians with non-trivial 7-torsion in Tate-Shafarevich groups.

Demonstrated visibility of 7-torsion in an abelian three-fold.

Provided explicit examples in small conductor families.

Abstract

Let $C/\mathbb{Q}$ be a genus $2$ curve whose Jacobian $J/\mathbb{Q}$ has real multiplication by a quadratic order in which $7$ splits. We describe an algorithm which outputs twists of the Klein quartic curve which parametrise elliptic curves whose mod $7$ Galois representations are isomorphic to a sub-representation of the mod $7$ Galois representation attached to $J/\mathbb{Q}$. Applying this algorithm to genus $2$ curves of small conductor in families of Bending and Elkies--Kumar we exhibit a number of genus $2$ Jacobians whose Tate--Shafarevich groups (unconditionally) contain a non-trivial element of order $7$ which is visible in an abelian three-fold.