Kisin modules with descent data and parahoric local models

TL;DR

This paper constructs a moduli space of Kisin modules with descent data and fixed p-adic Hodge type, demonstrating its smooth equivalence to a local model and relating it to Galois deformation rings, advancing understanding of p-adic Hodge theory.

Contribution

It introduces a new moduli space of Kisin modules with descent data, linking it to local models and Galois deformation rings, and establishes stratification via Kottwitz-Rapoport strata.

Findings

The moduli space $Y^{d, au}$ is smoothly equivalent to a local model for certain reductive groups.

Construction of Kottwitz-Rapoport strata on the special fiber of the moduli space.

Relation established between the moduli space and potentially crystalline Galois deformation rings.

Abstract

We construct a moduli space $Y^{\mu, \tau}$ of Kisin modules with tame descent datum $\tau$ and with fixed $p$-adic Hodge type $\leq \mu$, 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.