Arthur Packets and the Ramanujan Conjecture

TL;DR

This paper demonstrates that under a specific aspect of Arthur's A-packet conjecture, certain automorphic representations are tempered at almost all places, advancing understanding of the Ramanujan conjecture for quasisplit groups.

Contribution

It proves that, assuming part of Arthur's A-packet conjecture, locally generic cuspidal automorphic representations are of Ramanujan type, linking A-packets to temperedness at almost all places.

Findings

Automorphic representations are tempered at almost all places under the conjecture.

Reduction of the problem to local Langlands L-packets components.

General solution to a local question about L-packet components.

Abstract

The purpose of this paper is to show that under a part of generalized Arthur's A-packet conjecture, locally generic cuspidal automorphic representations of a quasisplit group over a number field are of Ramanujan type, i.e., are tempered at almost all places. The A-packet conjecture allows one to reduce the problem to a special case of a general local question about the components of the corresponding Langlands L-packet which is then answered here in its generality.