A deformation problem for Galois representations over imaginary quadratic fields

Details A deformation problem for Galois representations over imaginary quadratic fields A deformation problem for Galois representations over imaginary quadratic fields

2007-12-29

arxiv.org/abs/0801.0091v2

T. Berger and K. Klosin, A deformation problem for Galois representations over imaginary quadratic fields, Journal de l'instit

Mathematics

Number Theory

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 ar

Scientific paper

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.

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