The Jack Daniels problem

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• 作者：John F.R. Duncan^a ; ^{john.duncan@case.edu" class="auth_mail" title="E-mail the corresponding author} ; Ken Ono^b ; ^{ono@mathcs.emory.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11F11 ; 11G05 ; 20C34
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：161
• 期：Complete
• 页码：230-239
• 全文大小：296 K

文摘

In 1975 Ogg offered a bottle of Jack Daniels for an explanation of the fact that the prime divisors of the order of the monster class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002048&_mathId=si1.gif&_user=111111111&_pii=S0022314X15002048&_rdoc=1&_issn=0022314X&md5=93738bdc3c6c55a7198d2f7f2718eb1a" title="Click to view the MathML source">Mclass="mathContainer hidden">class="mathCode">$\mathbb{M}$ are the primes p for which the characteristic p supersingular j  -invariants are all defined over class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002048&_mathId=si2.gif&_user=111111111&_pii=S0022314X15002048&_rdoc=1&_issn=0022314X&md5=062c93a25d670c0a5192a85b5ad583fd" title="Click to view the MathML source">F_pclass="mathContainer hidden">class="mathCode">${\mathbb{F}}_p$. This coincidence is often suggested as the first hint of monstrous moonshine, the deep unexpected interplay between the monster and modular functions. We revisit Ogg's problem, and we point out (using existing tools) that the moonshine functions for order p elements give the set of characteristic p supersingular j-invariants (apart from 0 and 1728). Furthermore, we discuss this coincidence of the two seemingly unrelated sets of primes using the first principles of moonshine.