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 Γ*0(4) are found to be either Fourier coefficients of a weakly holomorphic modular form of weight 3/2 or constantmultiples 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.
ASJC Scopus subject areas
- General Mathematics