Mod-5 Galois images from abelian surfaces

Abstract

For $J$ an abelian surface, the Galois representation ${\varrho }_{J,\ell }:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \mathrm{Aut}(J[\ell ])\simeq {\mathrm{GSp}}_4({\mathbb{F}}_{\ell })$ is typically surjective, with smaller images indicating extra arithmetic structure. It is already known how to probabilistically compute whether ${\rho }_{J,\ell }$ is surjective, and recent work by Chidambaram computes $\mathrm{im}{\rho }_{J,\ell }$ for $\ell =2,3$. We probabilistically compute ${\rho }_{J,5}$ for the Jacobians of 95% of genus 2 curves in the L-functions and Modular Forms Database (LMFDB) for which ${\rho }_{J,5}$ is not yet known. For the remaining Jacobians, we determine the order of the image and give a short list of candidate images.

arXiv commentco-author emails: [email protected] [email protected]