Mathieu moonshine and Siegel Modular Forms

A second-quantized version of Mathieu moonshine leads to product formulae for functions that are potentially genus-two Siegel Modular Forms analogous the Igusa Cusp Form. The modularity these do not follow in an obvious manner. For some conjugacy classes, but all, they match known modular forms. In this paper, we express all classes $M_{24}$ terms products standard This provides a new proof their modularity.