On the generic part of the cohomology of local and global Shimura varieties

TL;DR

This paper proves a vanishing theorem for the generic unramified cohomology of local Shimura varieties, offering an alternative method to existing results for unitary Shimura varieties, based on Fargues-Scholze's work.

Contribution

It introduces a new vanishing theorem for local Shimura varieties using Fargues-Scholze's framework, extending the understanding of cohomological properties.

Findings

Vanishing of the generic unramified cohomology for local Shimura varieties.

Provides an alternative proof to existing vanishing results for unitary Shimura varieties.

Enhances the theoretical understanding of Shimura variety cohomology.

Abstract

Using the work of Fargues-Scholze, we prove a vanishing theorem for the generic unramified part of the cohomology of local Shimura varieties of general linear groups. This gives an alternative approach to vanishing results of Caraiani-Scholze for the cohomology of unitary Shimura varieties.