On the irreducible components of some crystalline deformation rings

Abstract

We adapt a technique of Kisin to construct and study crystalline deformation rings of $G_K$ for a finite extension $K/\mathbb{Q}_p$. This is done by considering a moduli space of Breuil--Kisin modules, satisfying an additional Galois condition, over the universal deformation ring. For $K$ unramified over $\mathbb{Q}_p$ and Hodge--Tate weights in $[0,p]$, we study the geometry of this space. As a consequence we prove that, under a mild cyclotomic-freeness assumption, all crystalline representations of an unramified extension of $\mathbb{Q}_p$, with Hodge--Tate weights in $[0,p]$, are potentially diagonalisable.