Math. Ann. (2011) 349:675–703 DOI 10.1007/s00208-010-0540-4 Mathematische Annalen An R = T theorem for imaginary quadratic fields Tobias Berger · Krzysztof Klosin Received: 2 December 2009 / Revised: 30 April 2010 / Published online: 5 June 2010 © Springer-Verlag 2010 Abstract We prove the modularity of certain residually reducible p-adic Galois representations of an imaginary quadratic field assuming the uniqueness of the resid- ual representation. We obtain an R = T theorem using a new commutative algebra criterion that might be of independent interest. To apply the criterion, one needs to show that the quotient of the universal deformation ring R by its ideal of reducibility is cyclic Artinian of order no greater than the order of the congruence module T/ J , where J is an Eisenstein ideal in the local Hecke algebra T. The inequality is proven by applying the Main conjecture of Iwasawa Theory for Hecke characters and using a result of Berger [Compos Math 145(3):603–632, 2009]. This strengthens our pre- vious result [Berger and Klosin, J Inst Math Jussieu 8(4):669–692, 2009] to include the cases of an arbitrary p-adic valuation of the L -value, in particular, cases when R is not a discrete valuation ring. As a consequence we show that the Eisenstein ideal is principal and that T is a complete intersection. Mathematics Subject Classification (2000) 11F80 · 11F55 · 11R34 · 13H10 1 Introduction Let K be a number field and ρ : Gal( K / K ) → GL 2 ( Q p ) a continuous irreducible rep- resentation. It has been a subject of much interest and effort lately to determine which T. Berger (B ) Queens’ College, University of Cambridge, Cambridge CB3 9ET, UK e-mail: t.berger@dpmms.cam.ac.uk K. Klosin Department of Mathematics, Queens College, City University of New York, 65-30 Kissena Boulevard, Flushing, NY 11367, USA e-mail: klosin@math.utah.edu 123