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arXiv · 2010.10021

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

Alexandru Ghitza·Takuya Yamauchi
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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 $\bar{\rho}_{C,2}$.

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Alexandru Ghitza, Takuya Yamauchi. 2020-10-18. Automorphy of mod 2 Galois representations associated to certain genus 2 curves over totally real fields. https://arxiv.org/abs/2010.10021

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