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

TL;DR

This paper introduces an algorithm to compute the mod-3 Galois image of a principally polarized abelian surface over the rationals, expanding computational methods beyond elliptic curves.

Contribution

It develops a novel algorithm to determine the Galois image for abelian surfaces at prime 3, handling subgroup distinctions without needing endomorphism data.

Findings

Algorithm successfully computes Galois images for abelian surfaces.

Distinguishes Gassmann-equivalent subgroups for GSp(4, F_3).

No endomorphism knowledge required.

Abstract

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.