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

Ghitza, Alexandru; Yamauchi, Takuya

Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 2, pp. 637-660

Given a genus two hyperelliptic curve $C$ over a totally real field $F$, we show that the mod $2$ Galois representation ${\overline{\rho }}_{C,2}:\mathrm{Gal}(\overline{F}/F)\to {\mathrm{GSp}}_4\left({𝔽}_2\right)$ attached to $C$ is residually automorphic when the image of ${\overline{\rho }}_{C,2}$ is isomorphic to $S_5$ and it is also a transitive subgroup under a fixed isomorphism ${\mathrm{GSp}}_4\left({𝔽}_2\right)\simeq S_6$. More precisely, there exists a Hilbert–Siegel Hecke eigen cusp form $h$ on ${\mathrm{GSp}}_4\left({𝔸}_F\right)$ of parallel weight two whose mod $2$ Galois representation ${\overline{\rho }}_{h,2}$ is isomorphic to ${\overline{\rho }}_{C,2}$.

Soit $C$ une courbe hyperelliptique de genre $2$ sur un corps totalement réel $F$. Nous montrons que la représentation galoisienne modulo $2$, ${\overline{\rho }}_{C,2}:\mathrm{Gal}(\overline{F}/F)\to {\mathrm{GSp}}_4\left({𝔽}_2\right),$ associée à $C$ est résiduellement automorphe lorsque l’image de ${\overline{\rho }}_{C,2}$ est un groupe isomorphe à $S_5$ et transitif par rapport à un isomorphisme fixé ${\mathrm{GSp}}_4\left({𝔽}_2\right)\simeq S_6$. Plus précisément, il existe une forme parabolique $h$ de Hilbert–Siegel sur ${\mathrm{GSp}}_4\left({𝔸}_F\right)$ de poids parallèle $2$, propre pour les opérateurs de Hecke, dont la représentation galoisienne ${\overline{\rho }}_{h,2}$ est isomorphe à ${\overline{\rho }}_{C,2}$.

Classification : 11F, 11F33, 11F80
Keywords: genus 2 curves, mod 2 Galois representations, automorphy

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