A DEFORMATION PROBLEM FOR GALOIS REPRESENTATIONS OVER IMAGINARY QUADRATIC FIELDS

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 Calois 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 oil all L-value) for the universal deformation ring to be a discrete valuation ring and in that case we prove all R = T theorem. We also study reducible deformations and show that no minimal characteristic 0 reducible deformation exists.