# On Traces of Singular Moduli

## TL;DR

This paper reviews and extends work on traces of singular moduli, providing new proofs, algorithms for modular forms, and explicit constructions of Hauptmoduln related to the Monster group.

## Contribution

It offers detailed proofs, algorithms for Jacobi and modular forms, and explicit Hauptmodul constructions for primes dividing the Monster group.

## Key findings

- New proofs of existing results on traces of singular moduli
- Algorithms for computing spaces of Jacobi and weight 3/2 modular forms
- Explicit Rademacher series-based Hauptmodul constructions for specific primes

## Abstract

This article is an overview of Zagier's and Kim's work on traces of singular moduli. We give more detailed or new proofs to some of their results and also describe some algorithms to compute spaces of Jacobi forms and weight $3/2$ modular forms. Also, for $p$ a prime dividing the order of the Monster group, we give an explicit construction of the Hauptmodul $j_p$ of $\Gamma_0(p)^*$ using Rademacher series and Rademacher sums.