On the Discrete Groups of Mathieu Moonshine

TL;DR

This paper establishes a link between the levels of certain cusp forms and the order of elements in the Mathieu group, supporting conjectures about the groups involved in Mathieu moonshine.

Contribution

It proves a one-to-one correspondence between levels of cusp forms and Mathieu group element orders, providing a new characterization related to Mathieu moonshine.

Findings

Cusp form space is one-dimensional iff level equals an element order of Mathieu group.

Supports conjectural classification of groups in Mathieu moonshine.

Analogue of Ogg's observation for the Mathieu group context.

Abstract

We prove that a certain space of cusp forms for the Hecke congruence group of a given level is one-dimensional if and only if that level is the order of an element of the second largest Mathieu group. As such, our result furnishes a direct analogue of Ogg's observation that the normaliser of a Hecke congruence group of prime level has genus zero if and only if that prime divides the order of the Fischer-Griess monster group. The significance of the cusp forms under consideration is explained by the Rademacher sum construction of the McKay-Thompson series of Mathieu moonshine. Our result supports a conjectural characterisation of the discrete groups and multiplier systems arising in Mathieu moonshine.