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

Let G G be a split semisimple group over function field. We prove the temperedness at unramified places of automorphic representations , subject to local assumption one place, stronger than supercuspidality, and assuming existence cyclic base change with good properties. Our method relies on geometry B u n Subscript upper Bun encoding="application/x-tex">\operatorname {Bun}_G . It is independent work Lafforgue global Langlands correspondence.