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

Abstract: In this talk, I will explain my recent work with Tsuzuki Nobuo on computing mod $2$ Galois representations $\overline{\rho}_{\psi,2}:G_K:={\rm Gal}(\overline{K}/K)\longrightarrow {\rm GSp}_4(\F_2)$ 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 $f_\psi(x)=4x^5-5\psi x^4+1$. In the course of the computation, we observe that the image of such a mod $2$ representation is governed by reciprocity of $f_\psi(x)$ whose decomposition field is generically of type 5-th symmetric group $S_5$. When K=F is totally real field, we apply the modularity of 2-dimensional, totally odd Artin representations of ${\rm Gal}(\overline{F}/F)$ due to Shu Sasaki to obtain automorphy of $\overline{\rho}_{\psi,2}$ after a suitable (at most) quadratic base extension.

algebraic geometrynumber theoryrepresentation theory

Audience: researchers in the topic

Columbia Automorphic Forms and Arithmetic Seminar