Diophantine analysis and Arthur's trace formula

2025/09/26 by Lee, Yuchan · 2 citations
#11D45 #11F72 #11G35 #11S80 #14F22 #14G12 #20G30 #20G35 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2509.22314

Abstract

Let X be a G-homogeneous space over a number field k such that X≅ G_γ\backslash G. Here, G is a simply connected semisimple group over k and γ∈ G(k) whose centralizer G_γ is a maximal torus in G which is anisotropic over k. We formulate the asymptotic for the number of integral points on X bounded by a fixed norm T>0 as T→ ∞ in terms of κ-orbital integrals, which play a role in the stabilization of Arthur's trace formula. This formula coincides with the contribution of the stable conjugacy class of γ to the geometric side of the trace formula. As an application, we obtain an asymptotic formula for the number of n × n matrices over the ring of integers O_k whose characteristic polynomial equals a fixed irreducible polynomial χ(x) of degree n. This result generalizes a case studied by Eskin-Mozes-Shah (1996).

Citations

• Stable orbital integrals for classical Lie algebras and smooth integral models
• Orbital integrals and ideal class monoids for a Bass order
• Orbital integrals and normalizations of measures
• Orbital integrals and Dedekind zeta functions
• Counting Integral Points in Certain Homogeneous Spaces
• Effective equidistribution of S-integral points on symmetric varieties
• A Statement of the Fundamental Lemma
• Rational points of bounded height on Fano varieties

Cited by

• Distributions of Integral Points and Dedekind Zeta Values
• Counting integral points in homogeneous spaces over function fields

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