Generic smoothness for $G$-valued potentially semi-stable deformation rings

TL;DR

This paper generalizes Kisin's results on Galois deformation rings to arbitrary reductive groups, establishing their complete intersection property and analyzing the structure of related moduli spaces, including singularities and resolutions.

Contribution

It extends the understanding of Galois deformation rings to arbitrary reductive groups and provides explicit analysis and resolutions for moduli spaces when G=GL_n.

Findings

Deformation rings are complete intersections for arbitrary connected reductive groups.

The structure of the moduli space X_{φ,N} is explicitly described for G=GL_n.

X_{φ,N} has singular components for G=GL_3, with constructed resolutions.

Abstract

We extend Kisin's results on the structure of characteristic $0$ Galois deformation rings to deformation rings of Galois representations valued in arbitrary connected reductive groups $G$. In particular, we show that such Galois deformation rings are complete intersection. In addition, we study explicitly the structure of the moduli space $X_{\varphi,N}$ of (framed) $(\varphi,N)$-modules when $G=\GL_n$. We show that when $G=\GL_3$ and $K_0=\Q_p$, $X_{\varphi,N}$ has a singular component, and we construct a moduli-theoretic resolution of singularities.