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This is a Quantitative Study study.

September 20, 2025Open Access

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

Key Points

Non-trivial elements of order 7 exist in the Tate–Shafarevich groups of some genus 2 jacobians, indicating unique properties.
The algorithm generates twists of the Klein quartic curve related to genus 2 jacobians, revealing remarkable Galois representations.
Genus 2 curves with small conductor from Bending and Elkies–Kumar families demonstrate interesting order 7 torsion.
An abelian three-fold reveals the visibility of these non-trivial order 7 group elements, enhancing understanding of elliptic curves.

Abstract

Let C/ℚ be a genus 2 curve whose Jacobian J/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/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.

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