O'Nan moonshine and arithmetic
Abstract
Answering a question posed by Conway and Norton in their seminal 1979 paper on moonshine, we prove the existence of a graded infinitedimensional module for the sporadic simple group of O'Nan, for which the McKayThompson series are weight $3/2$ modular forms. The coefficients of these series may be expressed in terms of class numbers, traces of singular moduli, and central critical values of quadratic twists of weight 2 modular $L$functions. As a consequence, for primes $p$ dividing the order of the O'Nan group we obtain congruences between O'Nan group character values and class numbers, $p$parts of Selmer groups, and TateShafarevich groups of certain elliptic curves. This work represents the first example of moonshine involving arithmetic invariants of this type.
 Publication:

arXiv eprints
 Pub Date:
 February 2017
 arXiv:
 arXiv:1702.03516
 Bibcode:
 2017arXiv170203516D
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Representation Theory;
 11F22;
 11F37
 EPrint:
 40 pages, 12 tables. v3: The McKayThompson series are all modular in this version. A connection to the cohomology of the O'Nan group is explained. Some clarifications and corrections have been made based on referee advice. v4: Some further clarifications, corrections and edits. This is the final version, accepted for publication in the American Journal of Mathematics