# Number-theoretic generalization of the Monster denominator formula

## TL;DR

This paper generalizes the Monster denominator formula using number-theoretic methods and polar harmonic Maass forms, extending its applicability to all modular curves $X_0(N)$ and connecting to Ramanujan's formulas and Green's functions.

## Contribution

It introduces a number-theoretic generalization of the Monster denominator formula for all $X_0(N)$ modular curves using polar harmonic Maass forms.

## Key findings

- Generalization of the Monster denominator formula to all $X_0(N)$
- Introduction to polar harmonic Maass forms
- Applications to Ramanujan's formulas and Green's functions

## Abstract

The denominator formula for the Monster Lie algebra is the product expansion for the modular function $j(z)-j(\tau)$ in terms of the Hecke system of $\operatorname{SL}_2(\mathbb{Z})$-modular functions $j_n(\tau)$. This formula can be reformulated entirely number-theoretically. Namely, it is equivalent to the description of the generating function for the $j_n(z)$ as a weight 2 modular form in $\tau$ with a pole at $z$. Although these results rely on the fact that $X_0(1)$ has genus 0, here we obtain a generalization, framed in terms of polar harmonic Maass forms, for all of the $X_0(N)$ modular curves. In this survey we discuss this generalization, and we offer an introduction to the theory of polar harmonic Maass forms. We conclude with applications to formulas of Ramanujan and Green's functions.