The algebraic meaning of genus-zero

Abstract

The Conway--Norton conjectures unexpectedly related the Monster with certain special modular functions (Hauptmoduls). Their proof by Borcherds et al was remarkable for demonstrating the rich mathematics implicit there. Unfortunately Moonshine remained almost as mysterious after the proof as before. In particular, a computer check - as opposed to a general conceptual argument - was used to verify the Monster functions equal the appropriate modular functions. This, the so-called `conceptual gap', was eventually filled; we review the solution here. We conclude by speculating on the shape of a new proof of the Moonshine conjectures.