Cycles on Shimura varieties via geometric Satake

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.

Preliminary version. Comments welcome

Cited by in corpus (22)

• Geometrization of the local Langlands correspondence
• Coherent sheaves on the stack of Langlands parameters
• The intersection motive of the moduli stack of shtukas
• Twisted orbital integrals and irreducible components of affine Deligne-Lusztig varieties
• Affine Grassmannians and the geometric Satake in mixed characteristic
• On Tate conjecture for the special fibers of some unitary Shimura varieties
• Smoothness of cohomology sheaves of stacks of shtukas
• Categorical traces and a relative Lefschetz-Verdier formula
• EKOR strata for Shimura varieties with parahoric level structure
• On vector-valued twisted conjugate invariant functions on a group
• Tate motives on Witt vector affine flag varieties
• Semi-modules and irreducible components of affine Deligne-Lusztig varieties
• Geometric Satake, categorical traces, and arithmetic of Shimura varieties
• Cohomology with integral coefficients of stacks of shtukas
• On quaternionic unitary Rapoport-Zink spaces with parahoric level structures
• Eichler-Shimura Relations for Shimura Varieties of Hodge Type
• Automorphic vector bundles on the stack of $G$-zips
• Hecke eigensystems of automorphic forms (mod $p$) of Hodge type and algebraic modular forms
• Deligne-Lusztig varieties and basic EKOR strata
• Special cycles on orthogonal Shimura varieties
• An alternative construction of zip period maps for Shimura varieties
• Mod $p$ points on Shimura varieties of parahoric level (with an appendix by Rong Zhou)