An asymptotic formula for the number of integral matrices with a fixed characteristic polynomial via orbital integrals

Abstract

For an irreducible polynomial $χ(x)\in \mathcal{O}_k[x]$ of degree $n$, where $k$ is a number field and $\mathcal{O}_k$ its ring of integers, let $N(X, T)$ denote the number of $n \times n$ integral matrices whose characteristic polynomial is $χ(x)$, bounded by a positive real number $T$ with respect to a certain norm. In this paper, we provide an asymptotic formula for $N(X,T)$ as $T\to \infty$ in terms of the orbital integrals of $\mathfrak{gl}_n$. This result extends the work of A. Eskin, S. Mozes, and N. Shah \cite{EMS} (1996) to a broader setting, thereby further developing the generalization initiated by the second author in arXiv:2509.22314. Our approach is based on the interpretation of local Brauer evaluations for $X$ via local class field theory, and on the Langlands-Shelstad fundamental lemma for $\mathfrak{sl}_n$. In particular, we observe that local Brauer evaluations for $X$ determine local endoscopic data for $\mathrm{SL}_n$, suggesting a deeper conceptual connection between these two notions.