Number of cuspidal automorphic representations and Hitchin's moduli spaces

TL;DR

This paper connects the count of specific cuspidal automorphic representations over function fields to the geometry of Hitchin moduli stacks, providing new insights into their structure and associated trace formulas.

Contribution

It establishes a link between automorphic representation counts and Hitchin moduli stacks, including geometric analysis and vanishing results for trace formulas.

Findings

Count of cuspidal automorphic representations expressed via Hitchin moduli stacks

Proved vanishing results for a variant of the Arthur-Selberg trace formula

Analyzed the geometry of Hitchin moduli stacks in this context

Abstract

Let $F$ be the function field of a projective smooth geometrically connected curve $X$ defined over a finite field $\mathbb{F}_q$. Let $G$ be a split semisimple algebraic group over $\mathbb{F}_q$. Let $S$ be a non-empty finite set of points of $X$. We are interested in the number of $G$ cuspidal automorphic representations whose local behaviors in $S$ are prescribed. In this article, we consider those cuspidal automorphic representations whose local component at each $v\in S$ contains a fixed irreducible Deligne-Lusztig induced representation of a hyperspecial group. We express that the count in terms of groupoid cardinality of $\mathbb{F}_q$-points of Hitchin moduli stacks of groups associated with $G$. In the course of the proof, we study the geometry of Hitchin moduli stacks and prove some vanishing results on the geometric side of a variant of the Arthur-Selberg trace formula for test functions with small support.