With a jocund air, we present an observation on the first 24 coefficients of
the modular invariant and of the modular discriminant. The observation is
purely for the sake of entertainment and could be of some diversion to a
mathematical audience.Comment: 1 pag

He, Y-H.

McKay, J.M.

arXiv

City Research Online

He, Y-H. & McKay, J.M. (2016). Moonshine and the Meaning of Life. Eureka, City Research OnlineOriginal citation: He, Y-H. & McKay, J.M. (2016). Moonshine and the Meaning of Life. Eureka, Permanent City Research Online URL: http://openaccess.city.ac.uk/12860/ Copyright & reuseCity University London has developed City Research Online so that its users may access the research outputs of City University London's staff. Copyright © and Moral Rights for this paper are retained by the individual author(s) and/ or other copyright holders.  All material in City Research Online is checked for eligibility for copyright before being made available in the live archive. URLs from City Research Online may be freely distributed and linked to from other web pages. Versions of researchThe version in City Research Online may differ from the final published version. Users are advised to check the Permanent City Research Online URL above for the status of the paper.EnquiriesIf you have any enquiries about any aspect of City Research Online, or if you wish to make contact with the author(s) of this paper, please email the team at publications@city.ac.uk.Moonshine and the Meaning of LifeYang-Hui He1 & John McKay21 Dept. of Maths, City U., London, EC1V 0HB, UK;School of Physics, NanKai U., Tianjin, 300071, P.R. China;Merton College, University of Oxford, OX14JD, UKhey@maths.ox.ac.uk2 Department of Mathematics and Statistics,Concordia University, 1455 de Maisonneuve West,Montreal, Quebec, H3G 1M8, Canadamckay@encs.concordia.caThe elliptic modular function, j, invariant under PSL(2,Z), has Fourier expansionj(q) =E4(q)3∆(q)=∞∑m=−1cmqm =1q+ 744 + 196884q + 21493760q2 + . . . , (1)as z → i∞, where q = e2piiz is the nome for z. E4(z) = 1 + 240∞∑n=1σ3(n)qn is the theta series for the E8lattice, σ3(n) =∑d|nd3 and∆(q) = q∞∏n=1(1− qn)24 =∞∑m=1τmqm = q − 24q2 + 252q3 − 1472q4 + 4830q5 + . . . (2)is the modular discriminant [S]. There are two new congruencesOBSERVATIONS: • [JM](24∑m=1c2m)mod 70 ≡ 42 ; • [YHH](24∑m=1τ2m)mod 70 ≡ 42.The vector ω = (0, 1, 2, . . . , 24 : 70) lives in the Lorentzian lattice II25,1 in 26 dimensions as anisotropic Weyl vector [C], allowing us to construct the Leech lattice as ω⊥/ω. Watson’s [L, W] uniquenon-trivial solution ton∑i=1i2 = m2 is (n,m) = (24, 70).Indeed, the second author’s observation 35 years ago that196884 = 196883 + 1 (3)sparked the field of “Monstrous Moonshine” [B, CN], underlying so much mathematics and physics,relating, inter alia, modular functions, finite groups, lattices, conformal field theory, string theory andgravity (see [G] for a review of some of the vast subjects encompassed) in which the j-invariant and theLeech lattice are central. As we ponder the meaning of life, we should be aware of the prescient remarksof the author [A], Douglas Adams:“The Answer to the Great Question . . . is . . . Forty-two,” said Deep Thought, with infinitemajesty and calm.References[A] Douglas Adams, “The Hitchhiker’s Guide to the Galaxy”, London, 1979.[B] R. E. Borcherds, “Vertex algebras, Kac-Moody algebras, & the monster,” Proc. Nat. Acad. Sci. 83, 3068(1986); “Monstrous moonshine & monstrous Lie superalgebras”, Invent. Math. 109 (1992) 405 - 444.[C] J. H. Conway, “The automorphism group of the 26-dim even unimodular Lorentzian lattice”, J. Alg. 80,Vol. 1, (1983), 159 - 163.[CN] J. H. Conway and S. P. Norton, “Monstrous Moonshine,” Bull. LMS, 11, (1979), 308 - 339.[G] T. Gannon, “Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics”,CUP, 2006, ISBN 0-521-83531-3.[L] E. Lucas, “Question 1180,” Nouvelles Annales de Mathe´matiques., ser. 14 (1875), 336.[S] J-P. Serre, “A Course in Arithmetic”, GTM 7, Presses Universitaires de France, (1970), §VII.3.3, 4.5[W] G. N. Watson, “The problem of the square pyramid,” Messenger of Mathematics, 48, (1918-19), 1-22arXiv:1408.2083v1  [math.NT]  9 Aug 2014

Moonshine and the Meaning of Life

Yang-Hui He

John McKay

Crossref

Moonshine and the meaning of life

The subgroup structure of the finite classical groups has long been the subject of intensive investigation. We explain some of the current issues relating to the study of the maximal subgroups of classical groups

Magaard, Kay

University of Birmingham Research Portal

English

Some remarks on maximal subgroups of finite classical groups

Finite Simple Groups: Thirty Years of the
                    Atlas and Beyond

Martin Liebeck

Character ratios for finite groups of Lie
                    type, and applications

For a  nite group
G
, a
character  ratio
is a complex number of the form
 
(
x
)
 
(1)
,
where
x
2
G
and
 
is an irreducible character of
G
.  Upper bounds for absolute
values of character ratios, particularly for simple groups, have long been of interest,
for various reasons; these include applications to covering numbers, mixing times
of random walks, and the study of word maps.  In this article we shall survey some
results  on  character  ratios  for   nite  groups  of  Lie  type,  and  their  applications.
Character ratios for alternating and symmetric groups have been studied in great
depth also { see for example [
32, 33
] { culminating in the de nitive results and
applications to be found in [
20
]; but we shall not discuss these here.
It is not hard to see the connections between character ratios and group struc-
ture.  Here are three well known, elementary results illustrating these connections.
The  rst two go back to Frobenius.  Denote by Irr(
G
) the set of irreducible charac-
ters of
G

Liebeck, MW

Spiral - Imperial College Digital Repository

Character ratios for finite groups of Lie type, and applications

RWTH Publications

Finite simple groups: thirty years of the Atlas and beyond : International Conference Finite Simple Groups: Thirty Years of the Atlas and Beyond : celebrating the Atlases and honoring John Conway, November 2-5, 2015, Princeton University, Princeton, NJ

Kay Magaard

Some remarks on maximal subgroups of finite
                    classical groups

http://pure-oai.bham.ac.uk/ws/files/36953860/2016SurveyDec.pdf

Moonshine and the Meaning of Life

Abstract

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University of Birmingham Research Portal

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