On divisors of modular forms

The denominator formula for the Monster Lie algebra is the product expansion for the modular function J(z)−J(τ) given in terms of the Hecke system of SL2(Z)-modular functions jn(τ). 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 jn(z) as a weight 2 modular form with a pole at z. Although these results rely on the fact that X0(1) has genus 0, here we obtain a generalization, framed in terms of polar harmonic Maass forms, for all of the X0(N) modular curves. We use these functions to study divisors of modular forms.