The number of cuspidal representations over a function field and its behavior under base changes

Abstract

Let $X$ be a smooth projective curve over a finite field $\mathbb{F}_q$, $k$ be its function field, and $G$ be a simply connected almost simple split group over $\mathbb{F}_q$. We also write $G$ for its structure over $k$. We calculate the sum of multiplicities of all cuspidal representations of $G$ satisfying a given condition assuming the conjectural trace formula. We also observe how the sum changes if we replace $X$ by its base change $X\otimes_{\mathbb{F}_q}\mathbb{F}_{q^m}$.