Explicit $7$-torsion in the Tate-Shafarevich groups of genus $2$ Jacobians
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916962204909568 |
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| author | Frengley, Sam |
| author_facet | Frengley, Sam |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_11693 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Explicit $7$-torsion in the Tate-Shafarevich groups of genus $2$ Jacobians Frengley, Sam Number Theory 11G30 (Primary) 11G10, 14H40 (Secondary) 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. |
| title | Explicit $7$-torsion in the Tate-Shafarevich groups of genus $2$ Jacobians |
| topic | Number Theory 11G30 (Primary) 11G10, 14H40 (Secondary) |
| url | https://arxiv.org/abs/2405.11693 |