Kisin modules with descent data and parahoric local models

Abstract

We construct a moduli space $Y^{\mu ,\tau }$ of Kisin modules with tame descent datum $\tau$ and with fixed $p$-adic Hodge type $\le \mu$, for some finite extension $K/{\mathbb{Q}}_p$. We show that this space is smoothly equivalent to the local model for ${\mathrm{Res}}_{K/{\mathbb{Q}}_p}{\mathrm{GL}}_n$, cocharacter $\{\mu \}$, and parahoric level structure. We use this to construct the analogue of Kottwitz-Rapoport strata on the special fiber $Y^{\mu ,\tau }$ indexed by the $\mu$-admissible set. We also relate $Y^{\mu ,\tau }$ to potentially crystalline Galois deformation rings.

38 pages, final version, to appear in Ann. Sci. de l'ENS