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

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.