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

TL;DR

This paper proves temperedness of automorphic representations over function fields for split semisimple groups using geometric methods, under certain local assumptions and base change conditions, without relying on Lafforgue's global Langlands work.

Contribution

It establishes temperedness results for automorphic forms over function fields via geometric techniques, independent of Lafforgue's approach, under specific local and base change assumptions.

Findings

Proves temperedness at unramified places for automorphic representations.

Uses geometry of $ ext{Bun}_G$ to achieve results.

Requires a local assumption stronger than supercuspidality and cyclic base change.

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 $\operatorname{Bun}_G$. It is independent of the work of Lafforgue on the global Langlands correspondence.