Modularity of M24-Twisted Siegel Product Expansions

One can attach Siegel product expansions of degree $2$ to elliptic genera of symmetric powers of K3 surfaces. The most popular example is $1 / \chi_{10}$, where $\chi_{10}$ is the Igusa cusp form of weight $10$.  In the setting of conjectural $M_{24}$ Moonshine this is one out of many examples $1 / \Phi_g$, each of which arises from a conjugacy class $g$ in the Mathieu group $M_{24}$.  Concrete expressions for these product expansion were given by Cheng, and were further discussed recently by Cheng and Duncan.  Apart from very few cases modularity of these products could not yet been proved.

We revisit this setting and present results on modularity of $\Phi_g$ for more than half of all conjugacy classes.