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

Abstract: For an arbitrarily given irreducible polynomial $\chi(x) \in \mathbb{Z}[x]$ of degree $n$ with $\mathbb{Q}[x]/(\chi(x))$ totally real, let $N(X, T)$ be the number of $n \times n$ matrices over $\mathbb{Z}$ whose characteristic polynomial is $\chi(x)$, bounded by a positive number $T$ with respect to a certain norm. In this paper, we will provide an asymptotic formula for $N(X, T)$ as $T \to \infty$ in terms of the orbital integrals of $\mathfrak{gl}_n$. This generalizes the work of A. Eskin, S. Mozes, and N. Shah (1996) which assumed that $\mathbb{Z}[x]/(\chi(x))$ is the ring of integers. In addition, we will provide an asymptotic formula for $N(X, T)$, using the orbital integrals of $\mathfrak{gl}_n$, when $\mathbb{Q}$ is generalized to a totally real number field $k$ and when $n$ is a prime number. Here we need a mild restriction on splitness of $\chi(x)$ over $k_v$ at $p$-adic places $v$ of $k$ for $p \leq n$ when $k[x]/(\chi(x))$ is unramified Galois over $k$. Our method is based on the strong approximation property with Brauer-Manin obstruction on a variety, the formula for orbital integrals of $\mathfrak{gl}_n$, and the Langlands-Shelstad fundamental lemma for $\mathfrak{sl}_n$.