Two moonshines for $L_2(11)$ but none for $M_{12}$

TL;DR

This paper explores the existence of moonshine phenomena for the groups $M_{12}$ and $L_2(11)$, constructing related modular forms and proving properties of their coefficients, revealing moonshine for $L_2(11)$ but not for $M_{12}$.

Contribution

It constructs Jacobi and Siegel modular forms for specific groups, providing new insights into moonshine phenomena and group-theoretic properties of modular objects.

Findings

No moonshine for $M_{12}$; solutions do not yield unique Jacobi forms.

Existence of moonshine for two $L_2(11)$ subgroups of $M_{12}$.

Each $L_2(11)$ conjugacy class corresponds to a Borcherds-Kac-Moody Lie superalgebra.

Abstract

In this paper, we revisit an earlier conjecture by one of us that related conjugacy classes of $M_{12}$ to Jacobi forms of weight one and index zero. We construct Jacobi forms for all conjugacy classes of $M_{12}$ that are consistent with constraints from group theory as well as modularity. However, we obtain 1427 solutions that satisfy these constraints (to the order that we checked) and are unable to provide a unique Jacobi form. Nevertheless, as a consequence, we are able to provide a group theoretic proof of the evenness of the coefficients of all EOT Jacobi forms associated with conjugacy classes of $M_{12}:2 \subset M_{24}$. We show that there exists no solution where the Jacobi forms (for order 4/8 elements of $M_{12}$) transform with phases under the appropriate level. In the absence of a moonshine for $M_{12}$, we show that there exist moonshines for two distinct $L_2(11)$ sub-groups of the $M_{12}$. We construct Siegel modular forms for all $L_2(11)$ conjugacy classes and show that each of them arises as the denominator formula for a distinct Borcherds-Kac-Moody Lie superalgebra.