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Mathematics, Number Theory
Abstract:
Let $K$ be an unramified extension of $\mathbb{Q}_p$ and $\rho\colon G_K
\rightarrow \operatorname{GL}_n(\overline{\mathbb{Z}}_p)$ 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 $p$.
