On the rigid cohomology of certain Shimura varieties.

Abstract

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