Inertial and Hodge--Tate weights of crystalline representations

Abstract

Let $K$ be an unramified extension of ${\mathbb{Q}}_p$ and $\rho :G_K\to {\mathrm{GL}}_n({\overline{\mathbb{Z}}}_p)$ a crystalline representation. If the Hodge--Tate weights of $\rho$ differ by at most $p$ then we show that these weights are contained in a natural collection of weights depending only on the restriction to inertia of $\overline{\rho }=\rho {\otimes }_{{\overline{\mathbb{Z}}}_p}{\overline{\mathbb{F}}}_p$. Our methods involve the study of a full subcategory of $p$-torsion Breuil--Kisin modules which we view as extending Fontaine--Laffaille theory to filtrations of length $p$.