On the Ramanujan conjecture for automorphic forms over function fields I. Geometry

Abstract

Let G be a split semisimple group over a function field. We prove the temperedness at unramified places of automorphic representations of G, subject to a local assumption at one place, stronger than supercuspidality, and assuming the existence of cyclic base change with good properties. Our method relies on the geometry of BunG. It is independent of the work of Lafforgue on the global Langlands correspondence.