Automorphy of mod 2 Galois representations associated to certain genus 2 curves over totally real fields
Journal de théorie des nombres de Bordeaux, Volume 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 ρ ¯ C,2 :Gal(F ¯/F)GSp 4 (𝔽 2 ) attached to C is residually automorphic when the image of 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 ρ ¯ h,2 is isomorphic to ρ ¯ C,2 .

Soit C une courbe hyperelliptique de genre 2 sur un corps totalement réel F. Nous montrons que la représentation galoisienne modulo 2, ρ ¯ C,2 :Gal(F ¯/F)GSp 4 (𝔽 2 ), associée à C est résiduellement automorphe lorsque l’image de 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 ρ ¯ h,2 est isomorphe à ρ ¯ C,2 .

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DOI: 10.5802/jtnb.1291
Classification: 11F, 11F33, 11F80
Keywords: genus 2 curves, mod 2 Galois representations, automorphy

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
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {Automorphy of mod~2 {Galois} representations associated to certain genus 2 curves over totally real fields},
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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, Volume 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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