  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.
