Distributions of Integral Points and Dedekind Zeta Values

Abstract

Let $\mathcal{O}$ be the ring of integers for some number field $F$. Let $\chi(x)\in \mathcal{O}[x]$ be a regular monic polynomial of degree $n$. We study the asymptotic count of integral $n\times n$ matrices over $\mathcal{O}$ with the characteristic polynomial $\chi$ and bounded archimedean norm. Previous works establish such an asymptotic with a positive leading constant. Our main result determines this constant in terms of the leading Laurent coefficients at $s=1$ of Dedekind zeta functions attached to orders in $F[x]/(\chi(x))$. The proof combines a refinement of the equi-distribution property of orbits with a reformulation of the counting problem in terms of generalized $\kappa$-orbital integrals. These orbital integrals are then transferred by the endoscopic fundamental lemma and related to zeta functions of orders.