Abstract

In this paper, we revisit an earlier conjecture by one of us that related conjugacy classes of M12 to Jacobi forms of weight zero and index one. We construct Jacobi forms for all conjugacy classes of M12 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 M12:2⊂M24. We show that there exists no solution where the Jacobi forms (for order 4/8 elements of M12) transform with phases under the appropriate level. In the absence of a moonshine for M12, we show that there exist moonshines for two distinct L2(11) sub-groups of the M12. We construct Siegel modular forms for all L2(11) conjugacy classes and show that each of them arises as the denominator formula for a distinct Borcherds–Kac–Moody Lie superalgebra.

Highlights

  • Following the discovery of monstrous moonshine, came a moonshine for the largest sporadic Mathieu group, M24

  • Our results may be summarised as follows: 1. We study a conjecture 2.1 that implies a moonshine for M12 that provides Jacobi forms of weight zero and index 1 for every conjugacy class of M12

  • This result, albeit non-unique, is sufficient to show that all the Fourier coefficients of the EOT Jacobi forms for M24 conjugacy classes that reduce to conjugacy classes of M12 : 2 are even

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Summary

Introduction

Following the discovery of monstrous moonshine, came a moonshine for the largest sporadic Mathieu group, M24 This related ρ, a conjugacy class of M24, to a multiplicative eta product that we denote by ηρ via the map [1, 2]: N ρ = 1a12a2 · · · N aN −→ ηρ(τ ) := η(mτ )am. We find 1427 families of Jacobi forms that have a positive definite character expansion This result, albeit non-unique, is sufficient to show that all the Fourier coefficients of the EOT Jacobi forms for M24 conjugacy classes that reduce to conjugacy classes of M12 : 2 (a maximal sub-group of M24) are even. We construct Siegel modular forms for all conjugacy classes using a product formula that arises naturally as a consequence of L2(11) moonshine. Siegel modular form of weight k for conjugacy class ρ of M24 Siegel modular form of weight k for conjugacy class ρof M12 and L2(11)

The M12 conjecture
Checking the M12 conjecture
Non-negativity of coefficients in the character expansion
Modularity
There is no moonshine for M12
The multiplicative lift
Modularity by comparing with the additive lift
Modularity by comparing with a Borcherds formula
The Siegel modular form admits the following Fourier expansion:
Establishing the Weyl-Kac-Borcherds denominator formula
Concluding Remarks
Weight two modular forms
B D satisfying
Examples
B Computations for the Borcherds product formula
M12 and M12 : 2

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