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O'Nan moonshine and arithmetic

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Mertens,  Michael H.
Max Planck Institute for Mathematics, Max Planck Society;

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Citation

Duncan, J. F. R., Mertens, M. H., & Ono, K. (2021). O'Nan moonshine and arithmetic. American Journal of Mathematics, 143(4), 1115-1159. doi:10.1353/ajm.2021.0029.


Cite as: https://hdl.handle.net/21.11116/0000-0009-72A4-D
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.