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arXiv · 2502.09753

Gluing Genus 1 and Genus 2 Curves Along $\ell$-torsion

Pitchayut Saengrungkongka·Noah Walsh
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Abstract

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.

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Pitchayut Saengrungkongka, Noah Walsh. 2025-02-13. Gluing Genus 1 and Genus 2 Curves Along $\ell$-torsion. https://arxiv.org/abs/2502.09753

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