Purity Reigns Supreme

Results of several authors have, in the last 20 years or so, associated Galois representations of degree n with cuspidal automorphic representations of adelic GL(n), over totally real or CM fields, satisfying certain conditions. We prove that, even in the case when the natural n-dimensional Galois representations cannot be constructed, the automorphic representations satisfy the Ramanujan conjecture at unramified primes.