Pariah moonshine

• Type of Material: Article
• Authors: John Duncan, Emory University; Michael H. Mertens, Mathematisches Institut der Universität zu Köln; Ken Ono, Emory University
• Language: English
• Date: 2017-09-22
• Publisher: Nature Publishing Group: Nature Communications
• Publication Version: Final Published Version
• Copyright Statement: © The Author(s) 2017
• License: Creative Commons Attribution 4.0 International
• Final Published Version (URL): http://dx.doi.org/10.1038/s41467-017-00660-y
• Title of Journal or Parent Work: Nature Communications
• ISSN: 2041-1723
• Volume: 8
• Start Page: 670
• End Page: 670
• Grant/Funding Information: This research was supported by the Asa Griggs Candler Fund (K.O.), the Max-Planck-Institut für Mathematik in Bonn (M.H.M.), the U.S. National Science Foundation, DMS 1601306 (J.F.R.D. and K.O.), and the Simons Foundation, #316779 (J.D.).
• Keywords: Quantum physics; Pure mathematics
• Research Categories: Physics, General; Mathematics

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.