Local-global compatibility of mod $p$ Langlands program for certain Shimura varieties

TL;DR

This paper extends the local-global compatibility in the mod p Langlands program to higher-dimensional Shimura varieties, establishing new relations between Scholze's functor and cohomology, and addressing torsion classes under specific conditions.

Contribution

It generalizes previous compatibility results to higher dimensions and introduces a cuspidality criterion from type theory for Shimura varieties.

Findings

Established local-global compatibility for higher-dimensional Shimura varieties.

Proved a cuspidality criterion based on type theory.

Addressed compatibility for torsion classes with semisimple mod p Galois representations.

Abstract

We generalize the local-global compatibility result in arXiv:1506.04022 to higher dimensional cases, by examining the relation between Scholze's functor and cohomology of Kottwitz-Harris-Taylor type Shimura varieties. Along the way we prove a cuspidality criterion from type theory. We also deal with compatibility for torsion classes in the case of semisimple mod $p$ Galois representations with distinct irreducible components under certain flatness hypotheses.