Conway--Norton monstrous moonshine conjecture

Let $\mathbb{M}$ be the monster, and for each $g\in\mathbb{M}$ let $T_g$ be its monstrous McKay--Thompson series. A normalized principal modulus is a generator of the function field of a genus-zero modular curve with the standard normalization at infinity. Conway--Norton conjecture. For each $g\in\mathbb{M}$ there is a specific group $\Gamma_g<\operatorname{SL}_2(\mathbb{R})$ such that $T_g$ is the unique normalized principal modulus for $\Gamma_g$. This is the principal-modulus, or genus-zero, property at the heart of monstrous moonshine; the statement is known as a consequence of the monstrous moonshine theorem.