The Moduli Stack of Breuil-Kisin Modules with Descent Data for Reductive Groups

TL;DR

This paper constructs and analyzes a moduli stack of Breuil-Kisin modules with additional structure, proving it is a $p$-adic formal algebraic stack equivalent to a twisted Schubert variety, advancing the understanding of $p$-adic geometry related to reductive groups.

Contribution

It introduces the moduli stack of Breuil-Kisin $( ext{Gamma}, ext{hat}G)$-torsors with descent data and proves its equivalence to a twisted Schubert variety, extending the geometric framework in $p$-adic Hodge theory.

Findings

The moduli stack $ ext{Y}^{ ext{leq}\mu}$ is a $p$-adic formal algebraic stack.

$ ext{Y}^{ ext{leq}\mu}$ is smoothly equivalent to a twisted Schubert variety.

The construction generalizes previous work to reductive groups with descent data.

Abstract

We introduce and study the moduli stack $\mathcal{Y}$ of Breuil-Kisin modules with $\hat{G}$-structure and descent data, or Breuil-Kisin $(\Gamma,\hat{G})$-torsors for short. Specifically, for a dominant cocharacter $\mu$, we define the moduli stack $\mathcal{Y}^{\leq \mu}$ of Breuil-Kisin $(\Gamma,\hat{G})$-torsors with Hodge-Tate weights bounded by $\mu$. We prove that $\mathcal{Y}^{\leq \mu}$ is a $p$-adic formal algebraic stack, and show that it is smoothly equivalent to (the $p$-adic completion of) a twisted Schubert variety $\operatorname{Gr}^{\leq \mu}_{\mathcal{G}}$ in the sense of Pappas-Zhu. This is a reformatted and lightly edited version of the author's PhD thesis, submitted to Northwestern University in August 2024.