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

TL;DR

This paper develops a systematic method to find genus 3 curves over Q whose Jacobians are isogenous to the product of genus 1 and genus 2 Jacobians via $ ext{ell}$-torsion, extending previous work for $ ext{ell}=2$ to primes up to 13.

Contribution

It introduces an improved numerical gluing algorithm and identifies new genus 1 and 2 curve pairs for primes up to 13, expanding the understanding of Jacobian isogenies.

Findings

Successfully glued genus 1 and genus 2 curves along 13-torsion.

Found several candidate pairs for primes up to 13.

Enhanced the numerical gluing algorithm for better performance.

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.