On the effective version of Serre's open image theorem

Mayle, Jacob

doi:10.48550/arxiv.2109.08656

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2023

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“…The present goal is to give an effective version of the Chebotarev density theorem in the context of abelian surfaces. We will use a corollary of (4) that is noted in [MW21] which allows for the avoidance of a prescribed set of primes by taking a quadratic extension of K. We do this because we will take K = Q(A[ ]), and p being unramified in K is not sufficient to imply that p is a prime of good reduction for A. Lastly, we will use that by [Ser81,Proposition 6…”

Section: 2mentioning

confidence: 99%

Computing nonsurjective primes associated to Galois representations of genus $2$ curves

S.¹,

Brumer²,

Kim³

et al. 2023

Preprint

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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 for which the Galois action on -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 PSp 4 (F ), 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 on a publicly-accessible branch of the LMFDB.

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Section: 2mentioning

confidence: 99%