From the Monster to Thompson to O'Nan

TL;DR

This paper explores the deep connections between sporadic simple groups, elliptic curves, and moonshine phenomena, revealing new relationships involving the O'Nan group and umbral moonshine.

Contribution

It uncovers novel links between the O'Nan group and elliptic curves, extending moonshine theory beyond the monster and Thompson groups.

Findings

Established a connection between O'Nan group and elliptic curves over rationals

Linked umbral moonshine to sporadic groups and elliptic curves

Extended moonshine framework to include O'Nan and umbral cases

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.