Pariah moonshine

TL;DR

This paper uncovers the O'Nan pariah group as a source of hidden symmetry in number theory, linking it to quadratic forms, elliptic curves, and deep mathematical conjectures.

Contribution

It demonstrates that pariah groups, previously thought unrelated to natural phenomena, have significant roles in number theory and mathematical structures.

Findings

Revealed O'Nan pariah group's role in quadratic forms and elliptic curves

Proved new congruences for class numbers and elliptic curve groups

Established pariah groups' relevance in fundamental mathematical problems

Abstract

Finite simple groups are the building blocks of finite symmetry. The effort to classify them precipitated the discovery of new examples, including the monster, and six pariah groups which do not belong to any of the natural families, and are not involved in the monster. It also precipitated monstrous moonshine, which is an appearance of monster symmetry in number theory that catalysed developments in mathematics and physics. Forty years ago the pioneers of moonshine asked if there is anything similar for pariahs. Here we report on a solution to this problem that reveals the O'Nan pariah group as a source of hidden symmetry in quadratic forms and elliptic curves. Using this we prove congruences for class numbers, and Selmer groups and Tate--Shafarevich groups of elliptic curves. This demonstrates that pariah groups play a role in some of the deepest problems in mathematics, and represents an appearance of pariah groups in nature.