Traces of singular values of Hauptmoduln

Abstract

In an important paper, Zagier proved that certain half-integral weight modular forms are generating functions for traces of polynomials in the j-function. It turns out that Zagier's work makes it possible to algorithmically compute Hilbert class polynomials using a canonical family of modular forms of weight [Formula: see text]. We generalize these results and consider Hauptmoduln for levels 1, 2, 3, 5, 7, and 13. We show that traces of singular values of polynomials in Hauptmoduln are again described by coefficients of half-integral weight modular forms. This realization makes it possible to algorithmically compute class polynomials.