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 : G K : = Gal ( K ¯ / K ) → GSp 4 ( 𝔽 2 ) associated to the mirror motives of rank 4 with pure weight 3 coming from the Dwork quintic family X 0 5 + X 1 5 + X 2 5 + X 3 5 + X 4 5 - 5 ψ X 0 X 1 X 2 X 3 X 4 = 0 , ψ ∈ K 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 quadratic 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 GSp 4 ( 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 ψ ( x ) = 4 x 5 - 5 ψ x 4 + 1 , ψ