Deformations of polarized automorphic Galois representations and adjoint Selmer groups

TL;DR

This paper proves the vanishing of a specific Selmer group for automorphic Galois representations, leading to smoothness results for deformation rings, using advanced patching and deformation techniques.

Contribution

It establishes the vanishing of the geometric Bloch-Kato Selmer group for adjoint automorphic Galois representations and proves smoothness of universal deformation rings at certain points.

Findings

Vanishing of the geometric Bloch-Kato Selmer group for adjoint representations.

Formal smoothness of deformation rings at automorphic points.

Characterization of smooth points on local deformation rings.

Abstract

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 fibre of universal deformation rings. Along the way we give a characterization of smooth closed points on the generic fibre of Kisin's potentially semistable local deformation rings in terms of their Weil-Deligne representations.