Automorphy of mod 2 Galois representations associated to certain genus 2 curves over totally real fields
A. Ghitza, T. Yamauchi
Let $C$ be a genus two hyperelliptic curve over a totally real field $F$. We show that the mod $2$ Galois representation $\rho$ attached to $C$ is automorphic when its image is a transitive subgroup of $S_6$ isomorphic to $S_5$. To be more precise, there exists a Hilbert--Siegel Hecke eigen cusp form of genus 2 and parallel weight 2 whose mod $2$ Galois representation is isomorphic to $\rho$.