Automorphy of mod 2 Galois representations associated to the quintic Dwork family and reciprocity of some quintic trinomials
Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 1, pp. 253-293

In this paper, we determine mod $2$ Galois representations $\bar{\rho }_{\psi ,2}:G_K:=\operatorname{Gal}(\bar{K}/K)\rightarrow \operatorname{GSp}_4(\mathbb{F}_2)$ associated to the mirror motives of rank 4 with pure weight 3 coming from the Dwork quintic family

\[ X^5_0+X^5_1+X^5_2+X^5_3+X^5_4-5\psi X_0X_1X_2X_3X_4=0,\ \psi \in K \]

defined over a number field $K$ under the irreducibility condition of the quintic trinomial $f_\psi $ below. Applying this result, when $K=F$ is a totally real field, for some at most quadratic totally real extension $M/F$, we prove that $\bar{\rho }_{\psi ,2}|_{G_M}$ is associated to a Hilbert–Siegel modular Hecke eigen cusp form for $\operatorname{GSp}_4(\mathbb{A}_M)$ of parallel weight three.

In the course of the proof, we observe that the image of such a mod $2$ representation is governed by reciprocity of the quintic trinomial

\[ f_\psi (x)=4x^5-5\psi x^4+1,\ \psi \in K \]

whose decomposition field is generically of type 5-th symmetric group $S_5$. This enable us to use results on the modularity of 2-dimensional, totally odd Artin representations of $\operatorname{Gal}(\bar{F}/F)$ due to Shu Sasaki or Pilloni–Shu and several Langlands functorial lifts for Hilbert cusp forms. Then, it guarantees the existence of a desired Hilbert–Siegel modular cusp form of parallel weight three matching with the Hodge type of the compatible system in question. A twisted version is also discussed and it is related to general quintic trinomials.

Dans cet article, nous déterminons les représentations galoisiennes modulaires $\bar{\rho }_{\psi ,2}: G_K:=\operatorname{Gal}(\bar{K}/K)\rightarrow \operatorname{GSp}_4(\mathbb{F}_2)$ associées aux motifs miroir de rang $4$ et de poids pur $3$ provenant de la famille quintique de Dwork

\[ X^5_0+X^5_1+X^5_2+X^5_3+X^5_4-5\psi X_0X_1X_2X_3X_4=0,\ \psi \in K \]

définie sur un corps de nombres $K$, sous la condition d’irréductibilité du trinôme quintique $f_\psi $ ci-dessous.

En appliquant ce résultat à un corps totalement réel $K=F$, nous prouvons que pour une extension totalement réelle $M/F$ de degré au plus $2$, $\bar{\rho }_{\psi ,2}|_{G_M}$ est associée à une forme de Hilbert–Siegel parabolique sur $\operatorname{GSp}_4(\mathbb{A}_M)$ de poids parallèle $3$, propre pour les opérateurs de Hecke.

Chemin faisant, nous observons que l’image d’une telle représentation de caractéristique $2$ est gouvernée par les lois de réciprocité associées au trinôme quintique

\[ f_\psi (x)=4x^5-5\psi x^4+1,\ \psi \in K \]

dont le corps de décomposition est génériquement de groupe de Galois $S_5$. Cela nous permet d’utiliser les résultats de modularité des représentations d’Artin de dimension $2$ totalement impaires de $\operatorname{Gal}(\bar{F}/F)$ dus à Shu Sasaki ou Pilloni–Shu ainsi que des divers relèvements fonctoriels de Langlands pour les formes modulaires de Hilbert. Il en résulte l’existence d’une forme de Hilbert–Siegel parabolique, de poids parallèle $3$, correspondant au type de Hodge du système compatible en question. Une version tordue est également discutée ; elle est liée aux trinômes quintiques généraux.

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DOI : 10.5802/jtnb.1361
Classification : 11F, 11F33, 11F80
Keywords: the quintic Dwork family, mod 2 Galois representations
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Nobuo Tsuzuki; Takuya Yamauchi. Automorphy of mod 2 Galois representations associated to the quintic Dwork family and reciprocity of some quintic trinomials. Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 1, pp. 253-293. doi: 10.5802/jtnb.1361
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