Inertial and Hodge–Tate weights of crystalline representations

Abstract

AbstractLet K be an unramified extension of $${\mathbb {Q}}_p$$Qp and $$\rho :G_K \rightarrow {\text {GL}}_n(\overline{{\mathbb {Z}}}_p)$$ρ:GK→GLn(Z¯p) a crystalline representation. If the Hodge–Tate weights of $$\rho $$ρ 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 $${\overline{\rho }} = \rho \otimes _{\overline{{\mathbb {Z}}}_p} \overline{{\mathbb {F}}}_p$$ρ¯=ρ⊗Z¯pF¯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.