Harmonic

Quantum mechanics

Product (mathematics)

Pure mathematics

Modular elliptic curve

Operating system

Elliptic curve

Computer science

Modular design

Hecke operator

Modular form

Moduli

Physics

Theta function

Mathematics

Quarter period

Mathematical analysis

Geometry

Hecke algebra

Eisenstein series

Generalization

Algebra over a field

Modular curve

Advances in Mathematics

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.

On divisors of modular forms