Twisted traces of singular moduli of weakly holomorphic modular functions

TL;DR

This paper extends Zagier's results on the modularity of twisted traces of singular moduli from the full modular group to congruence subgroups, and explores related congruences for these traces.

Contribution

It generalizes the modularity results of twisted traces of singular moduli to (N) subgroups and investigates associated congruences.

Findings

Twisted traces form weakly holomorphic modular forms on (N).

Established congruences for twisted traces of singular moduli.

Extended Zagier's results to arbitrary congruence subgroups.

Abstract

Zagier proved that the generating series for the traces of singular moduli is a \textit{weakly holomorphic} modular form of weight 3/2 on $\Gamma_0(4)$. Bruinier and Funke extended the results of Zagier to modular curves of arbitrary genus. Zagier also showed that the twisted traces of singular moduli are generated by a weakly holomorphic modular form of weight 3/2. In this paper, we study the extension of Zagier's result for the twisted traces of singular moduli to congruence subgroups $\Gamma_0(N)$. As an application, we study congruences for the twisted traces of singular moduli of weakly holomorphic modular functions on $\Gamma_0(N)$.