Mod-5 Galois images from abelian surfaces

2025/10/07 by Aidan Hennessey, Hennessey, Aidan, Mathilde Kermorgant +3
Computer Science · Engineering · #Advanced Numerical Analysis Techniques #Digital Image Processing Techniques #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2510.06423

openalex publication_date 2025/10/07 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28

Abstract

For J an abelian surface, the Galois representation \varrho_{J, ℓ} : \rm Gal(ℚ/ℚ) → \rm Aut(J[ℓ]) ≃ \rm GSp₄(\mathbbF_ℓ) is typically surjective, with smaller images indicating extra arithmetic structure. It is already known how to probabilistically compute whether ρ_{J, ℓ} is surjective, and recent work by Chidambaram computes im ρ_{J, ℓ} for ℓ = 2, 3. We probabilistically compute ρ_{J, 5} for the Jacobians of 95% of genus 2 curves in the L-functions and Modular Forms Database (LMFDB) for which ρ_{J, 5} is not yet known. For the remaining Jacobians, we determine the order of the image and give a short list of candidate images.

Citations

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

Related

• Computing the mod-3 Galois image of a principally polarized abelian surface over the rationals
• An explicit Abelian surface with maximal Galois action
• Large Galois images for Jacobian varieties of genus 3 curves
• Computing nonsurjective primes associated to Galois representations of genus 2 curves
• On the possible images of the mod ell representations associated to\n elliptic curves over Q