Abstract

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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Additional Information

ISSN
1080-6377
Print ISSN
0002-9327
Pages
pp. 1115-1159
Launched on MUSE
2021-07-10
Open Access
No
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