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.

Let p be a prime. We construct and study integral and torsion invariants, such as integral and torsion Weil–Deligne representations, associated to potentially semi-stable representations and torsion potentially semi-stable representations respectively. As applications, we prove the compatibility between local Langlands correspondence and Fontaine's construction for Galois representations attached to Hilbert modular forms, and Néron–Ogg–Shafarevich criterion of finite level for potentially semi-stable representations.