Nearly Ordinary Galois Deformations over Arbitrary Number Fields

TL;DR

This paper investigates the structure of nearly ordinary Galois deformation spaces over arbitrary number fields, highlighting differences between totally real and other fields, and providing new results for Artin and imaginary quadratic cases.

Contribution

It offers new theoretical results on nearly ordinary deformations over arbitrary fields, including unconditional results for imaginary quadratic fields and conjectures for non-totally real fields.

Findings

Proves a general result for Artin representations conditional on Leopoldt's conjecture.

Establishes the existence of positive dimensional components in deformation spaces over imaginary quadratic fields.

Suggests differences in deformation space structures between totally real and other number fields.

Abstract

Let K be an arbitrary number field, and let rho: Gal(Kbar/K) -> 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(Qbar/Q) when n > 2.