Gluing Genus 1 and Genus 2 Curves Along $\ell$-torsion
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916876740722688 |
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| author | Saengrungkongka, Pitchayut Walsh, Noah |
| author_facet | Saengrungkongka, Pitchayut Walsh, Noah |
| contents | Let $Y$ be a genus $2$ curve over $\mathbb Q$. We provide a method to systematically search for possible candidates of a prime $\ell\geq 3$ and a genus $1$ curve $X$ for which there exists a genus $3$ curve $Z$ over $\mathbb Q$ whose Jacobian is, up to quadratic twist, $(\ell, \ell, \ell)$-isogenous to the product of Jacobians of $X$ and $Y$, building on the work by Hanselman, Schiavone, and Sijsling for $\ell=2$. We find several such pairs $(X,Y)$ for prime $\ell$ up to $13$. We also improve their numerical gluing algorithm, allowing us to successfully glue genus $1$ and genus $2$ curves along their $13$-torsion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09753 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gluing Genus 1 and Genus 2 Curves Along $\ell$-torsion Saengrungkongka, Pitchayut Walsh, Noah Number Theory Algebraic Geometry 11G30, 14H25, 14H40 Let $Y$ be a genus $2$ curve over $\mathbb Q$. We provide a method to systematically search for possible candidates of a prime $\ell\geq 3$ and a genus $1$ curve $X$ for which there exists a genus $3$ curve $Z$ over $\mathbb Q$ whose Jacobian is, up to quadratic twist, $(\ell, \ell, \ell)$-isogenous to the product of Jacobians of $X$ and $Y$, building on the work by Hanselman, Schiavone, and Sijsling for $\ell=2$. We find several such pairs $(X,Y)$ for prime $\ell$ up to $13$. We also improve their numerical gluing algorithm, allowing us to successfully glue genus $1$ and genus $2$ curves along their $13$-torsion. |
| title | Gluing Genus 1 and Genus 2 Curves Along $\ell$-torsion |
| topic | Number Theory Algebraic Geometry 11G30, 14H25, 14H40 |
| url | https://arxiv.org/abs/2502.09753 |