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 ρ_ψ,2 : GK := Gal(K/K) −→ GSp₄(F2) associated to the mirror motives of rank 4 with pure weight 3 coming from the Dwork quintic family formula presented defined over a number field K under the irreducibility condition of the quintic trinomial f_ψ 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 ρ_ψ,2|G_M is associated to a Hilbert-Siegel modular Hecke eigen cusp form for GSp₄(AM) 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 formula presented whose decomposition field is generically of type 5-th symmetric group S5. This enable us to use results on the modularity of 2-dimensional, totally odd Artin representations of Gal(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.