• Corpus ID: 221265978

Automorphy of mod 2 Galois representations associated to the quintic Dwork family and reciprocity of some quintic trinomials

@article{Tsuzuki2020AutomorphyOM,
  title={Automorphy of mod 2 Galois representations associated to the quintic Dwork family and reciprocity of some quintic trinomials},
  author={Nobuo Tsuzuki and Takuya Yamauchi},
  journal={arXiv: Number Theory},
  year={2020}
}

• N. Tsuzuki, Takuya Yamauchi
• Published 22 August 2020
• Mathematics
• arXiv: Number Theory

In this paper, we determine mod $2$ Galois representations $\overline{\rho}_{\psi,2}:G_K:={\rm Gal}(\overline{K}/K)\longrightarrow {\rm GSp}_4(\mathbb{F}_2)$ 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 qaudratic totally real extension $M/F$, we prove that $\overline{\rho}_{\psi,2}|_{G_M}$ is associated to a Hilbert-Siegel modular Hecke eigen cusp form for ${\rm GSp}_4(\mathbb{A}_M)$ 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 ${\rm Gal}(\overline{F}/F)$ due to Shu Sasaki 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.