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

Abstract

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 p bigger than an effective bound, we prove that the Fp- and Zp-cohomology groups are concentrated in the middle degree, that the Zp-cohomology groups are free of p-torsion, and that every Fp-cohomology class lifts to a Zp-cohomology class.