2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/76524In this paper we complete the proof of Ryba's modular moonshine conjectures. We do this by applying Hodge theory to the cohomology of the monster Lie algebra over the ring of p-adic integers in order to calculate the Tate cohomology groups of elements of the monster acting on the monster vertex algebra.17 pages plain tex. To be published in Duke Math JQuantum AlgebraGroup Theorytext