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arXiv · 1702.03516

O'Nan moonshine and arithmetic

John F. R. Duncan·Michael H. Mertens·Ken Ono
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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 $3/2$ 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 Tate--Shafarevich groups of certain elliptic curves. This work represents the first example of moonshine involving arithmetic invariants of this type.

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John F. R. Duncan, Michael H. Mertens, Ken Ono. 2019-03-17. O'Nan moonshine and arithmetic. https://arxiv.org/abs/1702.03516

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