We prove the vanishing of the geometric Bloch–Kato Selmer group for the adjoint representation of a Galois representation associated to regular algebraic polarized cuspidal automorphic representations under an assumption on the residual image. Using this, we deduce that the localization and completion of a certain universal deformation ring for the residual representation at the characteristic zero point induced from the automorphic representation is formally smooth of the correct dimension. We do this by employing the Taylor–Wiles–Kisin patching method together with Kisin’s technique of analyzing the generic fiber of universal deformation rings. Along the way we give a characterization of smooth closed points on the generic fiber of Kisin’s potentially semistable local deformation rings in terms of their Weil–Deligne representations.
"Deformations of polarized automorphic Galois representations and adjoint Selmer groups." Duke Math. J. 165 (13) 2407 - 2460, 15 September 2016. https://doi.org/10.1215/00127094-3477342