Advances in Theoretical and Mathematical Physics

Volume 25 (2021)

Number 7

Module constructions for certain subgroups of the largest Mathieu group

Pages: 1703 – 1734

DOI: https://dx.doi.org/10.4310/ATMP.2021.v25.n7.a2

Author

Lea Beneish (Department of Mathematics, Emory University, Atlanta, Georgia, U.S.A.; and Department of Mathematics, University of California, Berkeley, Ca., U.S.A.)

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.

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Published 11 July 2022