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 GL2(AF) 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.

Citations