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

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 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),