2017/09/18 by John F. R. Duncan, Michael H. Mertens, Ken Ono · 2 citations
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Classification of finite simple groups #Combinatorics #Epistemology #Geometry #Group (periodic table) #Group of Lie type #Group theory #Law #Mathematics #Monster #Pariah group #Physics #Pure mathematics #Quantum mechanics #Simple (philosophy) #Simple group #Symmetry (geometry) #math.NT #math.RT #msc:11F22 #msc:11F37 #msc:11G05 #msc:11G40 #msc:20C34
paper · pdf · doi:10.1038/s41467-017-00660-y
published as Nature Communications 8(1) (2017) · 20 pages
openalex publication_date 2017/09/18 · arxiv created 2017/09/26 · arxiv updated 2017/09/27 · openalex created_date 2020/07/02 · openalex updated_date 2026/08/05
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.