Twisted traces of singular moduli of weakly holomorphic modular functions

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 $\Gamma_0(N)$.