Module constructions for certain subgroups of the largest Mathieu group

TL;DR

This paper constructs vertex operator algebra modules for specific subgroups of M24, linking them to meromorphic Jacobi forms and Mathieu moonshine, and provides explicit realizations of certain cusp forms and their arithmetic properties.

Contribution

It introduces new module constructions for subgroups of M24 that connect to meromorphic Jacobi forms and Mathieu moonshine, using a novel technique related to Conway moonshine.

Findings

Constructed vertex algebra modules for subgroups of M24.

Linked trace functions to meromorphic Jacobi forms and Mathieu moonshine.

Realized cusp forms with divisibility conditions on Jacobian point counts.

Abstract

For certain subgroups of $M_{24}$, we give vertex operator algebraic module constructions whose associated trace functions are meromorphic Jacobi forms. These meromorphic Jacobi forms are canonically associated to the mock modular forms of Mathieu moonshine. The construction is related to the Conway moonshine module and employs a technique introduced by Anagiannis--Cheng--Harrison. With this construction we are able to give concrete vertex algebraic realizations of certain cuspidal Hecke eigenforms of weight two. In particular, we give explicit realizations of trace functions whose integralities are equivalent to divisibility conditions on the number of $\mathbb{F}_p$ points on the Jacobians of modular curves.