Nearly Ordinary Galois Deformations over Arbitrary Number Fields

Abstract

Let (K) be an arbitrary number field, and let (\rho: Gal(K \bar/K) \rightarrow GL_2(E)) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of (\rho). When (K) is totally real and rho is modular, results of Hida imply that the nearly ordinary deformation space associated to rho 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 (\rho) is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of Leopoldt's 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 functorality", as well as some remarks on how our methods should apply to n-dimensional representations of Gal((Q \bar/Q)) when (n \lt 2).