Pariah moonshine

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.

20 pages

References in corpus (2)

• Multiplicative reduction and the cyclotomic main conjecture for ${\mathrm{GL}}_2$
• Fricke Lie algebras and the genus zero property in Moonshine

Cited by in corpus (4)

• Conformal Field Theories with Sporadic Group Symmetry
• Self-Dual Vertex Operator Superalgebras and Superconformal Field Theory
• On Weierstrass mock modular forms and a dimension formula for certain vertex operator algebras
• Hecke System of Harmonic Maass Functions and Applications to Modular Curves of Higher Genera