Mod p classification of Shimura F-crystals

TL;DR

This paper classifies certain algebraic structures called Shimura F-crystals mod p over algebraically closed fields and explores their stratifications, extending previous foundational work with new Bruhat F-decomposition techniques.

Contribution

It introduces and studies Bruhat F-decompositions, generalizing classical Bruhat decompositions, to classify D-truncations of Shimura F-crystals and analyze their stratifications.

Findings

Classification of D-truncations mod p of Shimura F-crystals

Development of Bruhat F-decompositions as a key tool

Extension of previous stratification results in algebraic geometry

Abstract

Let $k$ be an algebraically closed field of positive characteristic $p$. We first classify the $D$-truncations mod $p$ of Shimura $F$-crystals over $k$ and then we study stratifications defined by inner isomorphism classes of these $D$-truncations. This generalizes previous works of Kraft, Ekedahl, Oort, Moonen, and Wedhorn. As a main tool we introduce and study Bruhat $F$-decompositions; they generalize the combined form of Steinberg theorem and of classical Bruhat decompositions for reductive groups over $k$.