Automorphic realization of residual Galois representations (Q983895)

Summary: We show that it is possible in rather general situations to obtain a finite-dimensional modular representation \(\rho \) of the Galois group of a number field \(F\) as a constituent of one of the modular Galois representations attached to automorphic representations of a general linear group over \(F\), provided one works ``potentially.'' The proof is based on a close study of the monodromy of the Dwork family of Calabi-Yau hypersurfaces; this in turn makes use of properties of rigid local systems and the classification of irreducible subgroups of finite classical groups with certain sorts of generators.