Mathematische Annalen

, Volume 349, Issue 3, pp 675–703 | Cite as

An R = T theorem for imaginary quadratic fields

  • Tobias BergerEmail author
  • Krzysztof Klosin


We prove the modularity of certain residually reducible p-adic Galois representations of an imaginary quadratic field assuming the uniqueness of the residual 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 previous 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 


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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Queens’ College, University of CambridgeCambridgeUK
  2. 2.Department of MathematicsQueens College, City University of New YorkFlushingUSA

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