Distributions of Integral Points and Dedekind Zeta Values
Li Cai, Taiwang Deng
TL;DR
The paper analyzes the asymptotic distribution of integral matrices with a fixed irreducible characteristic polynomial over a number field. It develops an orbit-counting framework that expresses the leading constant $C(χ)$ in terms of Dedekind-type zeta values of orders $R=\mathcal{O}[x]/(χ)$, via equidistribution of orbits, Brauer-Manin obstructions, Tate-Nakayama duality, and endoscopic transfer. A central mechanism is relating κ-twisted orbital integrals to endoscopic orbital integrals and to residues of zeta functions of orders, enabling a sum over cyclic unramified intermediate fields $E/F$ with explicit weights. The result provides an explicit, arithmetic-density formula for the leading term of the distribution, connecting counting problems to the analytic behavior of zeta functions of orders and to the beyond-endoscopy philosophy for trace formulas.
Abstract
We study the distribution of integral matrices with a fixed characteristic polynomial. When the polynomial is irreducible, we determine the leading term of the distribution in terms of zeta functions of orders. The proof is based on an equidistribution property of orbits, a reformulation of the distribution into orbital integrals via the strong approximation with Brauer-Manin obstruction and the Tate-Nakayama duality, endoscopic fundamental lemma, and a critical input that links orbital integrals to residues of zeta functions of orders.