Automorphic Representations with Prescribed Ramification for Unitary Groups

Let F be a totally real number field, n a prime integer, and G a unitary group of rank n defined over F that is compact at infinity. We derive an asymptotic formula for the number of automorphic representations of G whose factors at finitely many places are prescribed up to inertia. These factors are specified by local inertial types, and the proof relies crucially on a bound on the traces of these types, which we establish first.