A deformation problem for Galois representations over imaginary quadratic fields
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.
Cite
@article{arxiv.0801.0091,
title = {A deformation problem for Galois representations over imaginary quadratic fields},
author = {Tobias Berger and Krzysztof Klosin},
journal= {arXiv preprint arXiv:0801.0091},
year = {2010}
}
Comments
22 pages; v2: added section 5.3 (gives a criterion for the univ. def. ring to be a dvr). A slightly modified version of the article published in J. Inst. Math. Jussieu. A related but stronger result is available at the authors' webpages - see e.g. http://www.math.utah.edu/~klosin and use the link "An R=T theorem for imaginary quadratic fields" (published version to appear in Math. Annalen)