Explicit $7$-torsion in the Tate–Shafarevich groups of genus $2$ Jacobians
Sam Frengley1
1 School of Mathematics, University of Bristol, Bristol, BS8 1UG, United Kingdom
Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 2, pp. 727-746

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.

Soit $C/\mathbb{Q}$ une courbe de genre $2$ dont la jacobienne $J/\mathbb{Q}$ a une multiplication réelle par un ordre quadratique dans lequel $7$ se décompose. Nous décrivons un algorithme qui produit une tordue galoisienne de la quartique de Klein qui paramètrise les courbes elliptiques dont la représentation galoisienne modulo 7 est isomorphe à une sous-représentation de la représentation galoisienne modulo $7$ associée à $J/\mathbb{Q}$. En appliquant cet algorithme aux courbes de genre $2$ de petit conducteur dans les familles de Bending et Elkies–Kumar nous donnons des exemples de courbes de genre $2$ dont les groupes de Tate–Shafarevich contiennent (inconditionellement) un élément non trivial d’ordre $7$ visible dans une variété abélienne de dimension $3$.

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DOI : 10.5802/jtnb.1340
Classification : 11G30, 11G10, 14H40
Keywords: Tate–Shafarevich group, Visibility, Modular curves, Galois representations

Sam Frengley 1

1 School of Mathematics, University of Bristol, Bristol, BS8 1UG, United Kingdom
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Explicit $7$-torsion in the {Tate{\textendash}Shafarevich} groups of genus $2$ {Jacobians}},
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Sam Frengley. Explicit $7$-torsion in the Tate–Shafarevich groups of genus $2$ Jacobians. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 2, pp. 727-746. doi: 10.5802/jtnb.1340

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