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

Alexandru Ghitza; Takuya Yamauchi

doi:10.5802/jtnb.1291

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

Alexandru Ghitza ¹ ; Takuya Yamauchi ²

¹ School of Mathematics and Statistics University of Melbourne Parkville, VIC 3010, Australia
² Mathematical Inst. Tohoku Univ. 6-3,Aoba, Aramaki, Aoba-Ku Sendai 980-8578, Japan

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

Résumé
Abstract

Soit $C$ une courbe hyperelliptique de genre $2$ sur un corps totalement réel $F$ . Nous montrons que la représentation galoisienne modulo $2$ , ${\bar{ρ}}_{C, 2} : Gal (\bar{F} / F) \to {GSp}_{4} (𝔽_{2}),$ associée à $C$ est résiduellement automorphe lorsque l’image de ${\bar{ρ}}_{C, 2}$ est un groupe isomorphe à $S_{5}$ et transitif par rapport à un isomorphisme fixé ${GSp}_{4} (𝔽_{2}) ≃ S_{6}$ . Plus précisément, il existe une forme parabolique $h$ de Hilbert–Siegel sur ${GSp}_{4} (𝔸_{F})$ de poids parallèle $2$ , propre pour les opérateurs de Hecke, dont la représentation galoisienne ${\bar{ρ}}_{h, 2}$ est isomorphe à ${\bar{ρ}}_{C, 2}$ .

Given a genus two hyperelliptic curve $C$ over a totally real field $F$ , we show that the mod $2$ Galois representation ${\bar{ρ}}_{C, 2} : Gal (\bar{F} / F) \to {GSp}_{4} (𝔽_{2})$ attached to $C$ is residually automorphic when the image of ${\bar{ρ}}_{C, 2}$ is isomorphic to $S_{5}$ and it is also a transitive subgroup under a fixed isomorphism ${GSp}_{4} (𝔽_{2}) ≃ S_{6}$ . More precisely, there exists a Hilbert–Siegel Hecke eigen cusp form $h$ on ${GSp}_{4} (𝔸_{F})$ of parallel weight two whose mod $2$ Galois representation ${\bar{ρ}}_{h, 2}$ is isomorphic to ${\bar{ρ}}_{C, 2}$ .

Reçu le : 2022-12-27
Révisé le : 2023-06-29
Accepté le : 2023-07-12
Publié le : 2024-11-13

DOI : 10.5802/jtnb.1291

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

Affiliations des auteurs :

Alexandru Ghitza ¹ ; Takuya Yamauchi ²

¹ School of Mathematics and Statistics University of Melbourne Parkville, VIC 3010, Australia
² Mathematical Inst. Tohoku Univ. 6-3,Aoba, Aramaki, Aoba-Ku Sendai 980-8578, Japan

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Alexandru Ghitza and Takuya Yamauchi},
     title = {Automorphy of mod~2 {Galois} representations associated to certain genus 2 curves over totally real fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {637--660},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {36},
     number = {2},
     year = {2024},
     doi = {10.5802/jtnb.1291},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1291/}
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Alexandru Ghitza; Takuya Yamauchi. Automorphy of mod 2 Galois representations associated to certain genus 2 curves over totally real fields. Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 2, pp. 637-660. doi : 10.5802/jtnb.1291. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1291/

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