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A deformation problem for Galois representations over imaginary quadratic fields

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  30 January 2009

Tobias Berger  and
Krzysztof Klosin
Tobias Berger
Affiliation:
University of Cambridge, Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Cambridge CB3 0WB, United Kingdom ([email protected])
Krzysztof Klosin
Affiliation:
Cornell University, Department of Mathematics, 310 Malott Hall, Ithaca, NY 14853-4201, USA ([email protected])
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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.

Keywords

Galois deformationsautomorphic formsimaginary quadratic fieldsmodularity

MSC classification

Secondary: 11F80: Galois representations 11F55: Other groups and their modular and automorphic forms (several variables)
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 8 , Issue 4 , October 2009 , pp. 669 - 692
Copyright
Copyright © Cambridge University Press 2009

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