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

2025/02/04 by Chidambaram, Shiva
#11F80 (Primary) 11G10 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2502.02044

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 ℚ. Conjugacy class distribution of subgroups of GSp(4,\mathbbF₃) is a key ingredient. While this ingredient is feasible to compute for GSp(4,\mathbbF_ℓ) for any small prime ℓ, distinguishing Gassmann-equivalent subgroups is a delicate problem. We accomplish it for ℓ = 3 using several techniques. The algorithm does not require the knowledge of endomorphisms.

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