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

Abstract

Let $C$ be a genus two hyperelliptic curve over a totally real field $F$. We show that the mod 2 Galois representation ${\bar{\rho }}_{C,2}:\mathrm{Gal}(\bar{F}/F)\to {\mathrm{GSp}}_4({\mathbb{F}}_2)$ 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 ${\bar{\rho }}_{C,2}$.

17 pages, 1 figure, 1 table. arXiv admin note: text overlap with arXiv:2008.09852

References in corpus (2)

• A database of genus 2 curves over the rational numbers
• On the images of Galois representations attached to low weight Siegel modular forms