A guide to the reduction modulo p of Shimura varieties

TL;DR

This paper reviews methods and results concerning the reduction modulo p of Shimura varieties with parahoric level structure, focusing on local and global theories, and proposing related conjectures.

Contribution

It provides a comprehensive overview of the local and global aspects of Shimura varieties reduction modulo p, including new conjectures on their point structures.

Findings

Explanation of parahoric subgroups and affine Deligne-Lusztig varieties

Formulation of conjectures on points in the reduction modulo p

Connection between local models and global structures

Abstract

This is a report on results and methods in the reduction modulo p of Shimura varieties with parahoric level structure. In the first part, the local theory, we explain the concepts of parahoric subgroups, of the mu-admissible and mu-permissible subsets of the Iwahori-Weyl group, of the corresponding union of affine Deligne-Lusztig varieties and of local models. In the second part, the global theory, we use these concepts to formulate conjectures on the points in the reduction modulo p of Shimura varieties with parahoric level structure.