Computing nonsurjective primes associated to Galois representations of genus 2 curves

2023/01/05 by Banwait, Barinder S., Brumer, Armand, Kim, Hyun Jong +4 · 1 citation
#11F80(primary) #11G10 #11Y16 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2301.02222

Abstract

For a genus 2 curve C over ℚ whose Jacobian A admits only trivial geometric endomorphisms, Serre's open image theorem for abelian surfaces asserts that there are only finitely many primes ℓ for which the Galois action on ℓ-torsion points of A is not maximal. Building on work of Dieulefait, we give a practical algorithm to compute this finite set. The key inputs are Mitchell's classification of maximal subgroups of PSp₄(\mathbbF_ℓ), sampling of the characteristic polynomials of Frobenius, and the Khare--Wintenberger modularity theorem. The algorithm has been submitted for integration into Sage, executed on all of the genus~2 curves with trivial endomorphism ring in the LMFDB, and the results incorporated into the homepage of each such curve.

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• Mod-5 Galois images from abelian surfaces

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