Cheng–Duncan conjecture on Mathieu moonshine Siegel modular forms

Let $M_{24}$ be the Mathieu group, let $g$ be a conjugacy class of $M_{24}$, and let $\Phi_g(\tau,z,\omega)$ be the formal infinite product defined by

$$\Phi_g(\tau,z,\omega)=q\zeta s\prod_{(n,r,m)>0}\exp\left(-\sum_{a=1}^{\infty}\frac{c_{g^a}(4nm-r^2)}{a}\left(q^n\zeta^r s^m\right)^a\right),$$

where $q=e^{2\pi i\tau}$, $\zeta=e^{2\pi iz}$, $s=e^{2\pi i\omega}$, and $(n,r,m)>0$ means that either $m>0$, or $m=0$ and $n>0$, or $m=n=0$ and $r<0$. For the integer $N_g$ associated to $g$, define

\Gamma_0^{(2)}(N_g)=\left\{\begin{psmallmatrix} A & B \\ C & D \end{psmallmatrix}\in\operatorname{Sp}_4(\mathbb Z): C=0\pmod{N_g}\right\}.

Cheng–Duncan conjecture. For each conjugacy class $g$ of $M_{24}$, the product $\Phi_g$ defines a Siegel modular form with some multiplier on the congruence subgroup $\Gamma_0^{(2)}(N_g)$.

The conjecture concerns the modularity of the products arising from Mathieu moonshine and would support an underlying $M_{24}$ symmetry and a general theory of twisted $\frac{1}{4}$-BPS spectra. It was subsequently resolved, so the conjectured modular-form property is established.