Automorphic realization of residual Galois representations

  • Michael Harris

    Centre Mathématiques de Jussieu, Paris, France
  • Nicholas M. Katz

    Princeton University, USA
  • Robert M. Guralnick

    University of Southern California, Los Angeles, United States


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

DOI 10.4171/JEMS/219