Abstract
We introduce a new method of proof for R = T theorems in the residually reducible case. We study the crystalline universal deformation ring R (and its ideal of reducibility I) of a mod p Galois representation ρ 0 of dimension n whose semisimplification is the direct sum of two absolutely irreducible mutually non-isomorphic constituents ρ 1 and ρ 2. Under some assumptions on Selmer groups associated with ρ 1 and ρ 2 we show that R/I is cyclic and often finite. Using ideas and results of (but somewhat different assumptions from) Bellaïche and Chenevier we prove that I is principal for essentially self-dual representations and deduce statements about the structure of R. Using a new commutative algebra criterion we show that given enough information on the Hecke side one gets an R = T-theorem. We then apply the technique to modularity problems for 2-dimensional representations over an imaginary quadratic field and a 4-dimensional representation over Q.
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Berger, T., Klosin, K. On deformation rings of residually reducible Galois representations and R = T theorems. Math. Ann. 355, 481–518 (2013). https://doi.org/10.1007/s00208-012-0793-1
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DOI: https://doi.org/10.1007/s00208-012-0793-1