Vanishing theorems for torsion automorphic sheaves on compact PEL-type Shimura varieties

Abstract

Given a compact PEL-type Shimura variety, a sufficiently regular weight (defined by mild and effective conditions), and a prime number $p$ unramified in the linear data and larger than an effective bound given by the weight, we show that the (Betti) cohomology with ${\mathbb{Z}}_p$-coefficients of the given weight vanishes away from the middle degree, and hence has no $p$-torsion. We do not need any other assumption (such as ones on the images of the associated Galois representations).