  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.
