Distributions of Integral Points and Dedekind Zeta Values

TL;DR

This paper investigates the asymptotic distribution of integral matrices with a fixed characteristic polynomial over number fields, linking the leading constants to Dedekind zeta function coefficients, and employs advanced orbital integral techniques.

Contribution

It provides a precise formula for the asymptotic count of matrices with a given characteristic polynomial, connecting it to Dedekind zeta values and utilizing endoscopic transfer methods.

Findings

Derived explicit asymptotic formulas for matrix counts

Connected leading constants to Laurent coefficients of Dedekind zeta functions

Applied orbital integrals and endoscopic transfer techniques

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.