Asian Journal of Mathematics

Volume 26 (2022)

Number 2

Stratifications in good reductions of Shimura varieties of abelian type

Pages: 167 – 226

DOI: https://dx.doi.org/10.4310/AJM.2022.v26.n2.a2

Authors

Xu Shen (Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China; and University of the Chinese Academy of Sciences, Beijing, China)

Chao Zhang (Shing-Tung Yau Center of Southeast University, Sipailou, Nanjing, China)

Abstract

In this paper we study the geometry of good reductions of Shimura varieties of abelian type. More precisely, we construct the Newton stratification, Ekedahl–Oort stratification, and central leaves on the special fiber of a Shimura variety of abelian type at a good prime. We establish several basic properties of these stratifications, including the non-emptiness, closure relation and dimension formula, generalizing those previously known in the PEL and Hodge type cases. We also study the relations between these stratifications, both in general and in some special cases, such as those of fully Hodge–Newton decomposable type. We investigate the examples of quaternionic and orthogonal Shimura varieties in details.

Keywords

Shimura varieties, Newton stratification, Ekedahl–Oort stratification

2010 Mathematics Subject Classification

Primary 14G35. Secondary 11G18.

Received 14 March 2019

Accepted 11 October 2021

Published 6 March 2023