Monstrous Moonshine conjecture

Let $\mathbb{M}$ be the Monster finite simple group, and let $J(q)$ be the normalized Hauptmodul for $\operatorname{SL}(2,\mathbb{Z})$. Monstrous Moonshine conjecture. There exists a natural infinite-dimensional $\mathbb{Z}$-graded $\mathbb{M}$-module $W=\bigoplus_{n=-1}^{\infty}W_n$ such that

$$\sum_{n=-1}^{\infty}\dim(W_n)q^n=J(q),$$

and, for every $g\in\mathbb{M}$,

$$\sum_{n=-1}^{\infty}\operatorname{tr}\left(g|_{W_n}\right)q^n$$

is the Hauptmodul of a genus-zero function field arising from a suitable discrete subgroup of $\operatorname{SL}(2,\mathbb{R})$. The conjecture is a central result of Monstrous Moonshine and has been proved through the construction of the Moonshine vertex operator algebra and subsequent work.