Density theorems for GL(n)
Abstract: The generalized Ramanujan conjecture predicts that all cuspidal automorphic representations for GL(n) are tempered. A density theorem is a quantitative version of the statement that non-tempered representations become rarer the further their Langlands parameters at a given place are away from the unitary axis. In many cases this is a good substitute for the Ramanujan conjecture, in the same way as the Bombieri-Vinogradov theorem can be used as a substitute for the Riemann hypothesis for Dirichlet L-functions. In this talk we show how a relative trace formula together with new bounds for GL(n) Kloosterman sums can obtain strong density theorems, and we present some applications.