Traces of Singular Moduli of Arbitrary Level Modular Functions

Generalizing Zagier's work, Bruinier and Funke recently proved that for modular curves of arbitrary genus, the generating series for the traces of the CM values of a weakly holomorphic modular function is the holomorphic part of a harmonic weak Maass form of weight 3/2. The present article shows that by adding a suitable linear combination of weight 3/2 Eisenstein series, one can always obtain a generating series that is weakly holomorphic. In particular, the modular traces of a Hauptmodul on $\mathrm\Gamma_0^\ast(4)$ are found to be either Fourier coefficients of a weakly holomorphic modular form of weight 3/2 or constant multiples of class numbers. As an application, we obtain congruence properties for the traces of singular moduli of a weakly holomorphic modular function of arbitrary level.