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

Let G G be a split semisimple group over a function field. We prove the temperedness at unramified places of automorphic representations of G 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 Bun G \operatorname {Bun}_G . It is independent of the work of Lafforgue on the global Langlands correspondence.