On the rigid cohomology of certain Shimura varieties

Abstract

We construct the compatible system of l-adic representations associated to a regular algebraic cuspidal automorphic representation of GLn over a CM (or totally real) field and check local–global compatibility for the l-adic representation away from l and a finite number of rational primes above which the CM field or the automorphic representation ramifies. The main innovation is that we impose no self-duality hypothesis on the automorphic representation.