Berger, Tobias; Klosin, Krzysztof On deformation rings of residually reducible Galois representations and \(R = T\) theorems. (English) Zbl 1292.11065 Math. Ann. 355, No. 2, 481-518 (2013). The main goal of this well-written article is to provide a new way to prove \(R=T\) theorems for crystalline deformation rings of certain residually reducible Galois representations. This is accomplished by first developing a novel commutative algebra isomorphism criterion (Sections 2–4, especially Theorem 4.1) which reduces the proof of \(R=T\) theorems to the study of the reducible deformations. Then, Sections 5–8 develop a method for proving \(R=T\) theorems at the level of reducible deformations using congruences of modular forms and bounds on Selmer groups. This yields Theorems 8.5 and 8.6, which provide two sets of “numerical conditions” that imply an \(R=T\) theorem, the first of which is interpreted as saying that cases of the Bloch–Kato conjecture imply \(R=T\) theorems.The basic setup under consideration is that of the crystalline self-dual deformation ring \(R_\Sigma\) of a crystalline \(n\)-dimensional non-semisimple Galois representation \(\rho_0:G_F\rightarrow\mathrm{GL}_n(\mathbb F)\) whose semi-simplification is the sum of two distinct absolutely irreducible representations \(\rho_1\) and \(\rho_2\). Here, \(G_F\) is the absolute Galois group of the number field \(F\) and \(\mathbb F\) is a finite field of characteristic \(p>2\). Several others assumptions, outlined in Section 6.1, are imposed, most prominently Assumption 6.1. In particular, the universal crystalline deformation rings of \(\rho_1\) and \(\rho_2\) are assumed to be discrete valuation rings and a certain Selmer group for \(\mathrm{Hom}_{\mathbf F}(\rho_2,\rho_1)\) is assumed to be \(1\)-dimensional. Theorem 8.5 can then be interpreted as explaining how the \(p\)-part of the Bloch–Kato conjecture for \(\operatorname{Hom}(\tilde{\rho}_2,\tilde{\rho}_1)\) implies an \(R=T\) theorem for \(\rho_0\) (here \(\tilde{\rho}_i\) is the universal crystalline deformation of \(\rho_i\)). A key ingredient to the commutative algebra involved is that the ideal of reducibility of \(R_\Sigma\) is principal. Section 2 shows that if a pseudocharacter is essentially self-dual, then its ideal of reducibility is principal.Two applications of the method developed in this paper are included in the final two sections. Section 9 proves an \(R=T\) theorem in the case where \(\rho_0\) is a \(2\)-dimensional Galois representation of an imaginary quadratic field, expanding the previous results of the authors [Math. Ann. 349, No. 3, 675–703 (2011; Zbl 1220.11073)] where the ordinary deformation ring was considered. Then, in Section 10, whose main results remain conjectural, the authors discuss \(4\)-dimensional representations of \(G_{\mathbb Q}\) that are residually isomorphic to certain Yoshida lifts (so that the semi-simplification of the residual representation is essentially a direct sum of two representations attached to elliptic cusp forms). Reviewer: Robert Harron (Madison) Cited in 1 ReviewCited in 7 Documents MSC: 11F80 Galois representations 11F55 Other groups and their modular and automorphic forms (several variables) Keywords:Galois deformation; automorphic form; R=T theorem; Bloch-Kato conjecture Citations:Zbl 1220.11073 PDFBibTeX XMLCite \textit{T. Berger} and \textit{K. Klosin}, Math. 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