On Goren-Oort stratification for quaternionic Shimura varieties

TL;DR

This paper investigates the Goren-Oort stratification of quaternionic Shimura varieties over totally real fields where p is unramified, revealing that each stratum forms a projective line bundle over related Shimura varieties.

Contribution

It demonstrates that each Goren-Oort stratum is a $( ext{P}^1)^N$-bundle over other quaternionic Shimura varieties, providing a detailed geometric structure of these stratifications.

Findings

Each stratum is a $( ext{P}^1)^N$-bundle.

The structure relates stratifications to quaternionic Shimura varieties.

Clarifies geometric properties of special fibers in the unramified case.

Abstract

Let $F$ be a totally real field in which $p$ is unramified. We study the Goren-Oort stratification of the special fibers of quaternionic Shimura varieties over a place above $p$. We show that each stratum is a $(\mathbb{P}^1)^N$-bundle over other quaternionic Shimura varieties (for some appropriate $N$).