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