Journal of the Institute of Mathematics of Jussieu

Research Article

Nearly ordinary Galois deformations over arbitrary number fields

Frank Calegaria1 and Barry Mazura2

a1 Department of Mathematics, Northwestern University, Lunt Hall, 2033 Sheridan Road, Evanston, IL 60208-2730, USA (

a2 Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA (


Let K be an arbitrary number field, and let ρ : Gal($\math{\bar{K}}$/K) → GL2(E) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of ρ. When K is totally real and ρ is modular, results of Hida imply that the nearly ordinary deformation space associated to ρ contains a Zariski dense set of points corresponding to ‘automorphic’ Galois representations. We conjecture that if K is not totally real, then this is never the case, except in three exceptional cases, corresponding to: (1) ‘base change’, (2) ‘CM’ forms, and (3) ‘even’ representations. The latter case conjecturally can only occur if the image of ρ is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of the Leopoldt Conjecture. Second, when K is an imaginary quadratic field, we prove an unconditional result that implies the existence of ‘many’ positive-dimensional components (of certain deformation spaces) that do not contain infinitely many classical points. Also included are some speculative remarks about ‘p-adic functoriality’, as well as some remarks on how our methods should apply to n-dimensional representations of Gal($\math{\bar{\QQ}}$/xs211A) when n > 2.

(Received October 31 2007)

(Revised January 16 2008)

(Accepted January 17 2008)


  • Galois deformations;
  • automorphic forms;
  • Hida families;
  • eigenvarieties;
  • p-adic modular forms

AMS 2000 Mathematics subject classification

  • Primary 11F75;
  • 11F80