A deformation problem for Galois representations over imaginary quadratic fields

TL;DR

This paper proves modularity and deformation properties of certain Galois representations over imaginary quadratic fields, establishing conditions for universal deformation rings and an R=T theorem, advancing understanding in number theory and automorphic forms.

Contribution

It introduces new conditions for the uniqueness and existence of deformations of Galois representations over imaginary quadratic fields, including an R=T theorem under specific assumptions.

Findings

Proved modularity of residually reducible Galois representations.

Established conditions for universal deformation rings to be discrete valuation rings.

Showed no minimal characteristic 0 reducible deformation exists.

Abstract

We prove the modularity of minimally ramified ordinary residually reducible p-adic Galois representations of an imaginary quadratic field F under certain assumptions. We first exhibit conditions under which the residual representation is unique up to isomorphism. Then we prove the existence of deformations arising from cuspforms on GL_2(A_F) via the Galois representations constructed by Taylor et al. We establish a sufficient condition (in terms of the non-existence of certain field extensions which in many cases can be reduced to a condition on an L-value) for the universal deformation ring to be a discrete valuation ring and in that case we prove an R=T theorem. We also study reducible deformations and show that no minimal characteristic 0 reducible deformation exists.