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

TL;DR

The paper derives an asymptotic formula for the count $N(X,T)$ of integral matrices with a fixed irreducible characteristic polynomial $\chi$ over a totally real number field, expressing the main term in terms of orbital integrals on $\mathfrak{gl}_n$ and a finite product over local places. The authors integrate a local Brauer-evaluation framework with the Langlands–Shelstad endoscopic philosophy, showing how Brauer data encodes endoscopic information for $\SL_n$ and using the fundamental lemma to compute unramified local contributions. Under prime degree and total-reality hypotheses, the result generalizes Eskin–Mozes–Shah to broader settings and recovers the EMS formula when $k=\mathbb{Q}$ (and $\chi(0)=1$ in the SL setting). The work highlights deep connections between Brauer evaluations, endoscopy, and orbital integrals, providing a concrete counting mechanism for integral matrices with prescribed spectral data and offering explicit constants and orbital-integral expressions for the asymptotics.

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.

Theorems & Definitions (61)

• Proposition 1.1
• Remark 1.2
• Proposition 1.3
• Proposition 1.4
• Proposition 1.5
• Lemma 1.6: Lemmas \ref{['lem:charstofevalxi']}–\ref{['lem:endo']}
• Theorem 1.7: Theorem \ref{['thm:main_thm']}
• Remark 1.8
• Proposition 2.1
• proof