The residual monodromy for the Dwork family in even characteristic and its applications to Galois representations

We study the residual monodromy representations associated to the Dwork family in characteristic two. Various applications involving 2-adic and mod 2 Galois representations are discussed. Combining the author's previous work with Tsuzuki and recent results of Boxer, Calegari, Gee, and Pilloni, we also prove the automorphy of certain rank 4 symplectic motives over a totally real field, arising from the Dwork quintic family, under suitable conditions.