Number theoretic generalization of the Monster denominator formula

Abstract

The denominator formula for the Monster Lie algebra is the product expansion for the modular function $j(z) - j( au)$ in terms of the Hecke system of $SL2(Z)$-modular functions $j_n(z)$. 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 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 of recent work, 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.