Vanishing theorems for the mod $p$ cohomology of some simple Shimura varieties

TL;DR

This paper proves that the mod p cohomology of certain simple Shimura varieties vanishes outside a specific range or the middle degree, depending on the dimension, after localization at non-Eisenstein ideals.

Contribution

It establishes vanishing theorems for mod p cohomology of simple Shimura varieties, extending known results to broader cases and specific degrees.

Findings

Cohomology vanishes outside a certain nontrivial range after localization.

In low dimensions, cohomology vanishes outside the middle degree.

Results apply to Shimura varieties discussed in Harris-Taylor's work.

Abstract

We show that the mod $p$ cohomology of a simple Shimura variety treated in Harris-Taylor's book vanishes outside a certain nontrivial range after localizing at any non-Eisenstein ideal of the Hecke algebra. In cases of low dimensions, we show the vanishing outside the middle degree.