Given a compact PEL-type Shimura variety, a sufficiently regular weight (defined by mild and effective conditions), and a prime number unramified in the linear data and larger than an effective bound given by the weight, we show that the (Betti) cohomology with -coefficients of the given weight vanishes away from the middle degree, and hence has no -torsion. We do not need any other assumption (such as ones on the images of the associated Galois representations).
Kai-Wen Lan. Junecue Suh. "Vanishing theorems for torsion automorphic sheaves on compact PEL-type Shimura varieties." Duke Math. J. 161 (6) 1113 - 1170, 15 April 2012. https://doi.org/10.1215/00127094-1548452