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

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.

arXiv comment22 pages (v2) Minor changes; (v3) 41 pages, Major revision includes new results; (v4) Final version

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• Mathieu Moonshine and Siegel Modular Forms
• $\widehat{sl(2)}$ decomposition of denominator formulae of some BKM Lie superalgebras -- II