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Mod-5 Galois images from abelian surfaces

Aidan Hennessey, Mathilde Kermorgant, Andy Zhu

TL;DR

The paper tackles determining the mod-$5$ Galois image $ ho_{J,5}$ for genus $2$ Jacobians, which typically surjects onto ${\rm GSp}_4(\mathbb{F}_5)$ but can be smaller when extra endomorphisms appear. It introduces an algorithm that fuses local Frobenius data from many primes with global torsion information (rational and quadratic $5$-torsion) to produce a short list of at most eight equal-order subgroups that likely contain $ ho_{J,5}$. The authors prove a main bound ensuring the search yields a concise candidate set and apply the method to 3990 non-surjective genus $2$ curves in the LMFDB, resolving the precise image in the majority of End$(J)$ scenarios and providing detailed classifications by endomorphism algebra. The work advances practical computation of Galois images in higher genus and links image structure to endomorphism types, with public code enabling replication and extension. The results reveal that rational or quadratic $5$-torsion are the primary obstructions to surjectivity, with most non-surjective cases arising from these factors and a minority explained by isogenies or complex multiplication in the factors.

Abstract

For $J$ an abelian surface, the Galois representation $\varrho_{J, \ell} : {\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow {\rm Aut}(J[\ell]) \simeq {\rm GSp}_4(\mathbb{F}_\ell)$ is typically surjective, with smaller images indicating extra arithmetic structure. It is already known how to probabilistically compute whether $ρ_{J, \ell}$ is surjective, and recent work by Chidambaram computes $\operatorname{im} ρ_{J, \ell}$ for $\ell = 2, 3$. 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.

Mod-5 Galois images from abelian surfaces

TL;DR

The paper tackles determining the mod- Galois image for genus Jacobians, which typically surjects onto but can be smaller when extra endomorphisms appear. It introduces an algorithm that fuses local Frobenius data from many primes with global torsion information (rational and quadratic -torsion) to produce a short list of at most eight equal-order subgroups that likely contain . The authors prove a main bound ensuring the search yields a concise candidate set and apply the method to 3990 non-surjective genus curves in the LMFDB, resolving the precise image in the majority of End scenarios and providing detailed classifications by endomorphism algebra. The work advances practical computation of Galois images in higher genus and links image structure to endomorphism types, with public code enabling replication and extension. The results reveal that rational or quadratic -torsion are the primary obstructions to surjectivity, with most non-surjective cases arising from these factors and a minority explained by isogenies or complex multiplication in the factors.

Abstract

For an abelian surface, the Galois representation is typically surjective, with smaller images indicating extra arithmetic structure. It is already known how to probabilistically compute whether is surjective, and recent work by Chidambaram computes for . We probabilistically compute for the Jacobians of 95% of genus 2 curves in the L-functions and Modular Forms Database (LMFDB) for which is not yet known. For the remaining Jacobians, we determine the order of the image and give a short list of candidate images.

Paper Structure

This paper contains 21 sections, 19 theorems, 15 equations, 2 figures, 1 table.

Table of Contents

  1. Background
  2. The possible images of Galois
  3. Mumford Coordinates
  4. Frobenius Elements
  5. Height Bounds for Torsion Points
  6. The Algorithm
  7. Local Information
  8. Sampling Algorithm
  9. Distribution Matching
  10. Global Information
  11. Rational 5-torsion
  12. Quadratic 5-torsion
  13. Proof of Correctness
  14. Results
  15. Subgroups and labels
  16. ...and 6 more sections

Key Result

Theorem 1

Let $C/\mathbb{Q}$ be a genus 2 hyperelliptic curve with Jacobian $J$. There exists an effective constant $N$, depending only on $C$, such that there exists an algorithm sampling all primes in the range $[10000, N]$ that produces a list of at most eight equal-order subgroups containing the mod-$5$ i

Figures (2)

  • Figure 1: A lattice of common images of Galois
  • Figure 2: A breakdown of the results of Algorithm \ref{['alg: imager']} on typical curves.

Theorems & Definitions (58)

  • Definition 1: Mod-$\ell$ Galois representation from torsion
  • Theorem : Main Theorem
  • Theorem : Computational Results
  • Definition 1.1.1: General symplectic group, ${\rm GSp}_{2g}(F)$, similitude character
  • Definition 1.1.2: $(\Lambda)$-group eigenspace
  • Definition 1.2.1: Mumford coordinates
  • Remark 1.2.2
  • Remark 1.2.3
  • Definition 1.3.1: 5-torsion field
  • Remark 1.3.2
  • ...and 48 more