Automorphy of mod 2 Galois representations associated to certain genus 2 curves over totally real fields

TL;DR

This paper proves the automorphy of certain mod 2 Galois representations arising from genus 2 hyperelliptic curves over totally real fields, linking them to Hilbert--Siegel cusp forms under specific group conditions.

Contribution

It establishes automorphy for mod 2 Galois representations associated to genus 2 curves with specific image properties, extending the understanding of their automorphic nature.

Findings

The mod 2 Galois representation is automorphic when its image is isomorphic to S_5.

Such representations correspond to Hilbert--Siegel cusp forms of parallel weight two.

The result applies to curves over totally real fields with transitive GSp_4(F_2) images.

Abstract

Let $C$ be a genus two hyperelliptic curve over a totally real field $F$. We show that the mod 2 Galois representation $\bar{\rho}_{C,2}\colon\mathrm{Gal}(\bar{F}/F)\to \mathrm{GSp}_4(\mathbb{F}_2)$ attached to $C$ is automorphic when the image of $\bar{\rho}_{C,2}$ is isomorphic to $S_5$ and it is also a transitive subgroup under a fixed isomorphism $\mathrm{GSp}_2(\mathbb{F}_2)\cong S_6$. To be more precise, there exists a Hilbert--Siegel Hecke eigen cusp form on $\mathrm{GSp}_4(\mathbb{A}_F)$ of parallel weight two whose mod 2 Galois representation is isomorphic to $\bar{\rho}_{C,2}$.