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An R = T theorem for imaginary quadratic fields

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Abstract

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.

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Authors and Affiliations

  1. Queens’ College, University of Cambridge, Cambridge, CB3 9ET, UK

    Tobias Berger

  2. Department of Mathematics, Queens College, City University of New York, 65-30 Kissena Boulevard, Flushing, NY, 11367, USA

    Krzysztof Klosin

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  1. Tobias Berger
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  2. Krzysztof Klosin
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Correspondence to Tobias Berger.

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Berger, T., Klosin, K. An R = T theorem for imaginary quadratic fields. Math. Ann. 349, 675–703 (2011). https://doi.org/10.1007/s00208-010-0540-4

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