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

In this paper, we determine mod $2$ Galois representations $\bar{\rho}_{\psi,2}:G_K:={\rm Gal}(\bar{K}/K)\longrightarrow {\rm GSp}_4(\mathbb{F}_2)$ 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 $\bar{\rho}_{\psi,2}|_{G_M}$ 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 ${\rm Gal}(\bar{F}/F)$ 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.A twisted version is also discussed and it is related to general quintic trinomials.