Cycles on Shimura varieties via geometric Satake

TL;DR

This paper constructs geometric correspondences between mod p fibers of Shimura varieties, linking their structure to affine Deligne-Lusztig varieties, and reveals how these relate to Tate classes in cohomology under certain conditions.

Contribution

It introduces a new method to analyze Shimura varieties via geometric Satake and affine Deligne-Lusztig varieties, providing insights into their cohomological structure and Newton strata.

Findings

Description of fibers of correspondences between Shimura varieties.

Identification of irreducible components generating Tate classes.

Determination of irreducible components of affine Deligne-Lusztig varieties.

Abstract

We construct (cohomological) correspondences between mod $p$ fibers of different Shimura varieties and describe the fibers of these correspondences by studying irreducible components of affine Deligne-Lusztig varieties. In particular, in the case of correspondences from a Shimura set to a Shimura variety, we obtain a description of the basic Newton stratum of the latter, and show that the irreducible components of the basic Newton stratum generate all the Tate classes in the middle cohomology of the Shimura variety, under a certain genericity condition. Along the way, we also determine the set of irreducible components of the affine Deligne-Lusztig variety associated to an unramified twisted conjugacy class.