On Divisors of Modular Forms

TL;DR

This paper generalizes the denominator formula for the Monster Lie algebra to all modular curves $X_0(N)$ using polar harmonic Maass forms, enabling new insights into divisors of modular forms beyond genus 0 cases.

Contribution

It introduces a new framework using polar harmonic Maass forms to study divisors of modular forms on all $X_0(N)$ curves, extending previous genus 0 results.

Findings

Generalization of the denominator formula to all $X_0(N)$

Application of polar harmonic Maass forms in divisor analysis

New methods for studying modular form divisors

Abstract

The denominator formula for the Monster Lie algebra is the product expansion for the modular function $j(z)-j(\tau)$ given in terms of the Hecke system of $\operatorname{SL}_2(\mathbb Z)$-modular functions $j_n(\tau)$. It is prominent in Zagier's seminal paper on traces of singular moduli, and in the Duncan-Frenkel work on Moonshine. The formula is equivalent to the description of the generating function for the $j_n(z)$ as a weight 2 modular form 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. We use these functions to study divisors of modular forms.