Stratifications in the reduction of Shimura varieties

TL;DR

This paper studies four different stratifications of Shimura varieties in characteristic p, establishing axioms that imply their non-emptiness and closure properties, with applications to the Siegel case.

Contribution

It formulates a unifying axiomatic framework for stratifications of Shimura varieties and verifies these axioms in the Siegel case, advancing understanding of their geometric structure.

Findings

A set of axioms implies non-emptiness of stratifications.

Closure relations among stratifications are established.

Axioms are verified for Siegel Shimura varieties.

Abstract

In the paper four stratifications in the reduction modulo $p$ of a general Shimura variety are studied: the Newton stratification, the Kottwitz-Rapoport stratification, the Ekedahl-Oort stratification and the Ekedahl-Kottwitz-Oort-Rapoport stratification. We formulate a system of axioms and show that these imply non-emptiness statements and closure relation statements concerning these various stratifications. These axioms are satisfied in the Siegel case.