Monstrous Moonshine for integral group rings

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.