Distribution of cusp sections in the Hilbert modular orbifold

TL;DR

This paper investigates how cusp cross sections in Hilbert modular orbifolds distribute as the height parameter approaches zero, linking the distribution rate to the Riemann hypothesis for Dedekind zeta functions.

Contribution

It generalizes Zagier's method to Hilbert modular orbifolds and establishes a connection between distribution rates and the Riemann hypothesis for number fields.

Findings

m(q) distributes uniformly as q approaches zero

The rate of distribution relates to the Riemann hypothesis for Dedekind zeta functions

Provides a new perspective on the distribution of cusp sections in Hilbert modular orbifolds

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.