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

Abstract

Let $Y$ be a genus $2$ curve over $\mathbb{Q}$. We provide a method to systematically search for possible candidates of a prime $\ell \ge 3$ 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.

28 pages. Presented at the conference "LMFDB, Computation, and Number Theory" (LuCaNT), Jul 7-11, 2025, at ICERM