The number of cuspidal representations over a function field and its behavior under base changes
Takuro Fukayama
TL;DR
The paper investigates the number of cuspidal automorphic representations of simply connected split groups over function fields with prescribed local behavior, using a conjectural trace formula together with the global Langlands correspondence. It expresses the count as sums of Artin–Tate motive $L$-functions $L_S(M_G)$ attached to the group and its centralizers, and analyzes how these sums change under base change to $\,bF_{q^m}$, revealing Lefschetz-type dependence on $m$. In the $SL_n$ case, the sum reduces to a divisor-sum over conjugacy-type centralizers, yielding explicit formulas when $n$ is prime and establishing a Lefschetz-type description under base change. For $Sp_{2n}$, the authors perform detailed classifications and computations for $Sp_4$ and $Sp_6$, verifying Lefschetz-type behavior, and conjecture a general Lefschetz-type pattern for all $Sp_{2n}$, tying base-change arithmetic to motivic $L$-functions and the trace formula.
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 $X\otimes_{\mathbb{F}_q}\mathbb{F}_{q^m}$.