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 infinite-dimensional module for the sporadic simple group of O'Nan, for which the McKay--Thompson series are weight 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 -functions. As a consequence, for primes dividing the order of the O'Nan group we obtain congruences between O'Nan group character values and class numbers, -parts of Selmer groups, and Tate--Shafarevich groups of certain elliptic curves. This work represents the first example of moonshine involving arithmetic invariants of this type.
Cite
@article{arxiv.1702.03516,
title = {O'Nan moonshine and arithmetic},
author = {John F. R. Duncan and Michael H. Mertens and Ken Ono},
journal= {arXiv preprint arXiv:1702.03516},
year = {2019}
}
Comments
40 pages, 12 tables. v3: The McKay--Thompson 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