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Duncan, John F. R., Mertens, Michael H. and Ono, Ken (2017). Pariah moonshine. Nat. Commun., 8. LONDON: NATURE PUBLISHING GROUP. ISSN 2041-1723

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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.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Duncan, John F. R. UNSPECIFIED UNSPECIFIED UNSPECIFIED Mertens, Michael H. UNSPECIFIED UNSPECIFIED UNSPECIFIED Ono, Ken UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-217451
DOI:	10.1038/s41467-017-00660-y
Journal or Publication Title:	Nat. Commun.
Volume:	8
Date:	2017
Publisher:	NATURE PUBLISHING GROUP
Place of Publication:	LONDON
ISSN:	2041-1723
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language ELLIPTIC-CURVES; MONSTER Multiple languages Multidisciplinary Sciences Multiple languages
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/21745