On the Modular Completion of Certain Generating Functions

We complete several generating functions to non-holomorphic modular forms in two variables. For instance, we consider the generating function of a natural family of meromorphic modular forms of weight two. We then show that this generating series can be completed to a smooth, non-holomorphic modular form of weights $\frac 32$ and two. Moreover, it turns out that the same function is also a modular completion of the generating function of weakly holomorphic modular forms of weight $\frac 32$, which prominently appear in work of Zagier [ 27] on traces of singular moduli.

ARITHMETIC PROPERTIES OF TRACES OF SINGULAR MODULI ON CONGRUENCE SUBGROUPS

After Zagier proved that the traces of singular moduli are Fourier coefficients of a weakly holomorphic modular form, various arithmetic properties of the traces of singular values of modular functions mostly on the full modular group have been found. The purpose of this paper is to generalize the results for modular functions on congruence subgroups with arbitrary level.

The vanishing of Poincaré series

Every holomorphic modular form of weight k > 2 is a sum of Poincaré series; see, for example, Chapter 5 of (5). In particular, every cusp form of even weight k ≧ 4 for the full modular group Γ(1) is a linear combination over the complex field C of the Poincaré series.Here mis any positive integer, z ∈ H ={z ∈ C: Im z>0} andThe summation is over all matriceswith different second rows in the (homogeneous) modular group, i.e. in SL(2, Z).The factor ½ is introducted for convenience.