Computing the mod-3 Galois image of a principally polarized abelian surface over the rationals

A lot of work has gone into computing images of Galois representations coming from elliptic curves. This article presents an algorithm to determine the image of the mod-$3$ Galois representation associated to a principally polarized abelian surface over $\mathbb{Q}$. Conjugacy class distribution of subgroups of $\mathrm{GSp}(4,\mathbb{F}_3)$ is a key ingredient. While this ingredient is feasible to compute for $\mathrm{GSp}(4,\mathbb{F}_{\ell})$ for any small prime $\ell$, distinguishing Gassmann-equivalent subgroups is a delicate problem. We accomplish it for $\ell = 3$ using several techniques. The algorithm does not require the knowledge of endomorphisms.