Mathieu Moonshine and Siegel Modular Forms

TL;DR

This paper explores Mathieu moonshine's connection to genus-two Siegel Modular Forms, providing new product formulae and a proof of their modularity for all conjugacy classes of M24.

Contribution

It expresses product formulae for all M24 conjugacy classes in terms of standard modular forms, offering a new proof of their modularity.

Findings

Product formulae for all M24 conjugacy classes are expressed in terms of standard modular forms.

A new proof of the modularity of these functions is provided.

Some functions match known modular forms, but not all, indicating partial correspondence.

Abstract

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