A coarse geometric expansion of a variant of Arthur's truncated traces and some applications

Hongjie Yu

Vol. 321 (2022), No. 1, 193–237

DOI: 10.2140/pjm.2022.321.193

Abstract

Let $F$ be a global function field with constant field ${\mathbb{𝔽}}_q$. Let $G$ be a reductive group over ${\mathbb{𝔽}}_q$. We establish a variant of Arthur’s truncated kernel for $G$ and for its Lie algebra which generalises Arthur’s original construction. We establish a coarse geometric expansion for our variant truncation.

As applications, we consider some existence and uniqueness problems of some cuspidal automorphic representations for the functions field of the projective line ${\mathbb{P}}_{{\mathbb{𝔽}}_q}^1$ with two points of ramifications.

Keywords

Arthur–Selberg trace formula, cuspidal automorphic representations, global function fields

Mathematical Subject Classification

Primary: 11F70, 11R58

Milestones

Received: 14 March 2022

Revised: 24 June 2022

Accepted: 29 August 2022

Published: 3 March 2023