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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.