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.

arXiv comment12 pages. v3 fixed referencing error (article unchanged). v2 small changes (numbering updated to match published version). An example constructed in this paper has appeared in the appendix to arXiv:2312.07307

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