On cuspidality of global Arthur packets for symplectic groups

TL;DR

This paper investigates which global Arthur packets for symplectic groups lack cuspidal automorphic representations, enhancing understanding of automorphic spectra and their relation to the Ramanujan conjecture.

Contribution

It provides new criteria and insights into the cuspidality of global Arthur packets for symplectic groups, extending classical automorphic form theory.

Findings

Identification of conditions for non-cuspidal packets

Deeper understanding of local unramified components

Connections to the generalized Ramanujan problem

Abstract

In [Ar13], Arthur classifies the automorphic discrete spectrum of symplectic groups up to global Arthur packets, based on the theory of endoscopy. It is an interesting and basic question to ask: which global Arthur packets contain no cuspidal automorphic representations? The investigation on this question can be regarded as a further development of the topics originated from the classical theory of singular automorphic forms. The results obtained yield a better understanding of global Arthur packets and of the structure of local unramified components of the cuspidal spectrum, and hence are closely related to the generalized Ramanujan problem as posted by Sarnak in [Sar05].