G-Valued Crystalline Deformation Rings in the Fontaine-Laffaille Range

TL;DR

This paper investigates the structure of crystalline deformation rings for G-valued Galois representations, establishing conditions for formal smoothness, especially for exceptional groups, extending previous results beyond classical types.

Contribution

It introduces a root theoretic criterion ensuring the formal smoothness of G-valued crystalline deformation rings, including for exceptional groups, advancing the understanding of deformation theory in p-adic Hodge contexts.

Findings

Root theoretic condition for smoothness of deformation rings

First results for exceptional groups in this context

Improves upon all known results for non-type A classical groups

Abstract

Let $G$ be a split reductive group over the ring of integers in a $p$-adic field with residue field $\mathbf{F}$. Fix a representation $\overline{\rho}$ of the absolute Galois group of an unramified extension of $\mathbf{Q}_p$, valued in $G(\mathbf{F})$. We study the crystalline deformation ring for $\overline{\rho}$ with a fixed $p$-adic Hodge type that satisfies an analog of the Fontaine-Laffaille condition for $G$-valued representations. In particular, we give a root theoretic condition on the $p$-adic Hodge type which ensures that the crystalline deformation ring is formally smooth. Our result improves on all known results for classical groups not of type A and provides the first such results for exceptional groups.