Traces of singular values of Hauptmoduln

TL;DR

This paper generalizes Zagier's work on modular forms and traces of singular values to Hauptmoduln for specific levels, enabling algorithmic computation of class polynomials through half-integral weight modular forms.

Contribution

It extends the connection between traces of singular values and modular forms to Hauptmoduln at levels 1, 2, 3, 5, 7, and 13, facilitating new computational methods.

Findings

Traces of singular values are described by coefficients of half-integral weight modular forms.

The results enable algorithmic computation of class polynomials for multiple levels.

Generalization of Zagier's results to a broader class of modular functions.

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 $\frac{3}{2}$. We generalize these results and consider Haupmoduln for levels $1, 2, 3, 5, 7,$ and $13$. We show that traces of singular values of polynomials in Haupmoduln are again described by coefficients of half-integral weight modular forms. This realization makes it possible to algorithmically compute class polynomials.