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.Comment: 20 page
