Norton's generalized moonshine conjecture

Let $\mathbb{M}$ be the monster and let $V^{\natural}_g$ denote the unique simple $g$-twisted module. For commuting $g,h\in\mathbb{M}$, choose a lift $\tilde h$ of $h$ to $V^{\natural}_g$, and define the associated trace functions by

$$T_{(g,\tilde h)}(\tau)=\sum_n \operatorname{tr}(\tilde h\mid(V^{\natural}_g)_n)q^n.$$

Norton's generalized moonshine conjecture. There is an assignment of holomorphic functions $T_{(g,\tilde h)}:\mathbb{H}\to\mathbb{C}$ to every commuting pair $(g,h)$ in the monster satisfying: conjugacy invariance; modular covariance up to scalar; the assertion that the coefficient functions are characters of projective representations of the centralizer of $g$; the assertion that each function is constant or generates the function field of a genus-zero group $\Gamma_{(g,h)}<\operatorname{SL}_2(\mathbb{R})$; and the condition that it is a scalar multiple of $J$ exactly when $g=h=e$. Generalized moonshine extends monstrous moonshine to twisted sectors and commuting pairs; the full conjectural picture has been developed through later work on twisted modules and genus-zero properties.