Vanishing theorems for the mod $p$ cohomology of some simple Shimura varieties (1910.03147v3)

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.