Arithmetic Properties of Traces of Singular Moduli on Congruence Subgroups

TL;DR

This paper extends the study of arithmetic properties of traces of singular moduli from the full modular group to congruence subgroups, revealing new formulas, distribution patterns, and congruences.

Contribution

It generalizes known properties of traces of singular moduli from the full modular group to congruence subgroups, broadening understanding of their arithmetic nature.

Findings

Derived explicit formulas for traces on congruence subgroups

Analyzed distribution and duality properties of these traces

Established congruences for traces on $\Gamma_0(N)$

Abstract

After Zagier proved that the traces of singular moduli $j(z)$ are Fourier coefficients of a weakly holomorphic modular form, various properties of the traces of the singular values of modular functions mostly on the full modular group $PSL_2(\mathbb{Z})$ have been investigated such as their exact formulas, limiting distribution, duality, and congruences. The purpose of this paper is to generalize these arithmetic properties of traces of singular values of a weakly holomorphic modular function on the full modular group to those on a congruence subgroup $\Gamma_0(N)$.