We show that it is possible in rather general situations to obtain a ﬁnite-dimensional modular representation ρ 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.
Cite this article
Michael Harris, Nicholas M. Katz, Robert M. Guralnick, Automorphic realization of residual Galois representations. J. Eur. Math. Soc. 12 (2010), no. 4, pp. 915–937