Monstrous Moonshine for integral group rings

TL;DR

This paper generalizes Monstrous Moonshine by proposing a conjecture linking group ring representations to Hauptmodul functions, reducing it to a genus-zero problem and proving it for specific cyclic subgroups of the Monster.

Contribution

It introduces a broad conjecture connecting integral group rings and Moonshine modules, reducing it to a genus-zero problem and proving it for certain cyclic subgroups.

Findings

Proves the conjecture for cyclic subgroups generated by elements in class 4A.

Determines multiplicities in the decomposition of the integral Moonshine Module.

Establishes a connection between representation rings and Hauptmodul functions.

Abstract

We propose a conjecture that is a substantial generalization of the genus zero assertions in both Monstrous Moonshine and Modular Moonshine. Our conjecture essentially asserts that if we are given any homomorphism to the complex numbers from a representation ring of a group ring for a subgroup of the Monster, we obtain a Hauptmodul by applying this homomorphism to a self-dual integral form of the Moonshine module. We reduce this conjecture to the genus-zero problem for "quasi-replicable" functions, by applying Borcherds's integral form of the Goddard-Thorn no-ghost theorem together with some analysis of the Laplacian on an integral form of the Monster Lie algebra. We prove our conjecture for cyclic subgroups of the Monster generated by elements in class 4A, and we explicitly determine the multiplicities for a decomposition of the integral Moonshine Module into indecomposable modules of the integral group rings for these groups.