Automorphic Realization of Residual Galois Representations

The “Galois representations” of the title are modular representations ρ of the Galois groups of a number field F , and the ”automorphic realization” of the title refers to obtaining these representations as constituents of Galois representations attached to automorphic representations of general linear groups over F . The present article refines the moduli-theoretic arguments of [HST] to show that this is possible in rather general situations, provided one works “potentially,” replacing ρ by its restriction to a certain infinite class of Galois extensions F ′/F ; this class is sufficiently large that the restriction to the Galois group of F ′ can be assumed injective. In §1, we introduce the notion of potential stable automorphy of modular galois representations, and state a general result on the ubiquity of such representations. In §2 we state some rather precise grouptheoretic results on the monodromy of the Dwork family, strengthening the results of [HST], and use them to prove the general result of §1. In §3 we discuss variants and possible future applications of the general result. In §4 we prove the group-theoretic results stated in §2, as well as some supplements to those results. The techniques used in §4 are basd on (1), relating the monodromy of the Dwork family to a rigid local system, then exploiting properties of rigid local systems, and (2), applying results on the classification of irreducible subgroups of finite classical groups with certain sorts of generators.