2024/12/31 by Seongsoo Jeon, Jeon, Seongsu, Yuchan Lee +1 · 3 citations
Mathematics · #11F72 #11G35 #11R37 #11S80 #14F22 #14G12 #20G30 #20G35 #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT) #Random Matrices and Applications #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2501.00284
openalex publication_date 2024/12/31 · openalex created_date 2025/01/04 · openalex updated_date 2026/07/28
For an irreducible polynomial χ(x)∈ Ok[x] of degree n, where k is a number field and Ok its ring of integers, let N(X, T) denote the number of n × n integral matrices whose characteristic polynomial is χ(x), bounded by a positive real number T with respect to a certain norm. In this paper, we provide an asymptotic formula for N(X,T) as T→ ∞ in terms of the orbital integrals of \mathfrakgln. This result extends the work of A. Eskin, S. Mozes, and N. Shah \citeEMS (1996) to a broader setting, thereby further developing the generalization initiated by the second author in arXiv:2509.22314. Our approach is based on the interpretation of local Brauer evaluations for X via local class field theory, and on the Langlands-Shelstad fundamental lemma for \mathfraksln. In particular, we observe that local Brauer evaluations for X determine local endoscopic data for SLn, suggesting a deeper conceptual connection between these two notions.