From the Monster to Thompson to O'Nan

Abstract

The commencement of monstrous moonshine is a connection between the largest sporadic simple group---the monster---and complex elliptic curves. Here we explain how a closer look at this connection leads, via the Thompson group, to recently observed relationships between the non-monstrous sporadic simple group of O'Nan and certain families of elliptic curves defined over the rationals. We also describe umbral moonshine from this perspective.

21 pages, accepted for publication in Contemporary Mathematics, Proceedings of the Conference on Vertex Operator Algebras, Number Theory and Related Topics