On deformation rings of residually reducible Galois representations and R = T theorems

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 ρ0 of dimension n whose semisimplification is the direct sum of two absolutely irreducible mutually non-isomorphic constituents ρ1 and ρ2. Under some assumptions on Selmer groups associated with ρ1 and ρ2 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.