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

We consider the interior cohomology (and the Hodge graded pieces in the case of the de Rham realization) of general–not necessarily compact–PEL-type Shimura varieties with coefficients in the local systems corresponding to sufficiently regular algebraic representations of the associated reductive group. For primes pp bigger than an effective bound, we prove that the FpFp- and ZpZp-cohomology groups are concentrated in the middle degree, that the ZpZp-cohomology groups are free of pp-torsion, and that every FpFp-cohomology class lifts to a ZpZp-cohomology class.