Distribution of cusp sections in the Hilbert modular orbifold

2011/12/06 by Samuel Estala Arias, Arias, Samuel Estala
Mathematics · #11M36 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11M36

paper · pdf · doi:10.48550/arxiv.1112.1146

21 pages

arxiv created 2011/12/06 · openalex publication_date 2011/12/06 · arxiv updated 2011/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a number field, let M be the Hilbert modular orbifold of K, and let m(q) be the probability measure uniformly supported on the cusp cross sections of M at height q. We generalize a method of Zagier and show that m(q) distributes uniformly with respect to the normalized Haar measure m on M as q tends to zero, and relate the rate by which m(q) approaches m to the Riemann hypothesis for the Dedekind zeta function of K.

Related

• Arithmetic, geometry and dynamics in the unit tangent bundle of the modular orbifold
• Distribution of modular symbols for compact surfaces
• On the supremum of random cusp forms
• Distribution of integral Fourier Coefficients of a Modular Form of Half Integral Weight Modulo Primes
• On the slopes of the lattice of sections of Hermitian line bundles