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arXiv · 2510.06423

Mod-5 Galois images from abelian surfaces

Aidan Hennessey·Mathilde Kermorgant·Andy Zhu
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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 $\rho_{J, \ell}$ is surjective, and recent work by Chidambaram computes $\operatorname{im} \rho_{J, \ell}$ for $\ell = 2, 3$. We probabilistically compute $\rho_{J, 5}$ for the Jacobians of 95% of genus 2 curves in the L-functions and Modular Forms Database (LMFDB) for which $\rho_{J, 5}$ is not yet known. For the remaining Jacobians, we determine the order of the image and give a short list of candidate images.

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Aidan Hennessey, Mathilde Kermorgant, Andy Zhu. 2025-10-07. Mod-5 Galois images from abelian surfaces. https://arxiv.org/abs/2510.06423

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