Mod-5 Galois images from abelian surfaces

Abstract: For $J$ an abelian surface, the Galois representation $\varrho_{J, \ell} : {\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow {\rm Aut}(J[\ell]) \simeq {\rm GSp}4(\mathbb{F}\ell)$ 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 $\operatorname{im} \rho_{J, \ell}$ 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.