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

TL;DR

This paper develops a new variant of Arthur's truncated kernel for reductive groups over function fields, providing a coarse geometric expansion and applying it to automorphic representation problems with ramification.

Contribution

It introduces a generalized truncated kernel and geometric expansion for reductive groups over function fields, extending Arthur's original framework.

Findings

Established a variant of Arthur's truncated kernel for G and its Lie algebra.

Derived a coarse geometric expansion for the new truncation.

Applied the expansion to automorphic representation existence and uniqueness problems.

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 generalizes 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 $\mathbb{P}^1_{\mathbb{F}_q}$ with two points of ramifications.