Purity of level m stratifications

TL;DR

This paper proves the purity of level m stratifications in certain algebraic structures over fields of positive characteristic, with specific results depending on the prime characteristic and applications to Shimura varieties.

Contribution

It establishes the purity of level m stratifications for BT_m over fields of characteristic p, extending known results and applying techniques to Shimura varieties of Hodge type.

Findings

Purity holds for p ≥ 5, with the stratification being an affine immersion.

Purity for p = 2, 3 depends on a property of the p-torsion D_m[p].

All level m stratifications of Shimura varieties of Hodge type are pure for p ≥ 5.

Abstract

Let $k$ be a field of characteristic $p>0$. Let $D_m$ be a $\BT_m$ over $k$ (i.e., an $m$-truncated Barsotti--Tate group over $k$). Let $S$ be a\break $k$-scheme and let $X$ be a $\BT_m$ over $S$. Let $S_{D_m}(X)$ be the subscheme of $S$ which describes the locus where $X$ is locally for the fppf topology isomorphic to $D_m$. If $p\ge 5$, we show that $S_{D_m}(X)$ is pure in $S$ i.e., the immersion $S_{D_m}(X) \hookrightarrow S$ is affine. For $p\in\{2,3\}$, we prove purity if $D_m$ satisfies a certain property depending only on its $p$-torsion $D_m[p]$. For $p\ge 5$, we apply the developed techniques to show that all level $m$ stratifications associated to Shimura varieties of Hodge type are pure.