Monstrous moonshine

In mathematics, monstrous moonshine is a term devised by John Horton Conway and Simon P. Norton in 1979, used to describe the (then totally unexpected) connection between the monster group M and modular functions (particularly, the j function).
Specifically, Conway and Norton found that the Fourier expansion of j(τ) (OEIS A000521 (http://www.research.att.com/cgibin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A000521), with τ denoting the halfperiod ratio) could be expressed in terms of linear combinations of the dimensions of the irreducible representations of M (OEIS A001379 (http://www.research.att.com/cgibin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001379))
 <math>j(\tau) = \frac{1}{{q}} + 744 + 196884{q} + 21493760{q}^2 + 864299970{q}^3 + \cdots<math>
where <math>{q} = e^{2\pi i\tau}<math>, and
<math>1 \,\!<math> <math>= \,\!<math> <math>1 \,\!<math> <math>196884 \,\!<math> <math>= \,\!<math> <math>196883 + 1 \,\!<math> <math>21493760 \,\!<math> <math>= \,\!<math> <math>21296876 + 196883 + 1 \,\!<math> <math>864299970 \,\!<math> <math>= \,\!<math> <math>842609326 + 21296876 + 2\cdot 196883 + 2\cdot 1 \,\!<math> ...
Conway and Norton formulated conjectures concerning the functions <math>j_g({q})<math> obtained by replacing the traces on the identity by the traces on other elements g of M. The most striking part of these conjectures is that all these functions are genus zero. In other words, if G_{g} is the subgroup of SL_{2}(R) which fixes <math>j_g({q})<math>, then the quotient of the upper half of the complex plane by G_{g} is a sphere with a finite number of points removed, corresponding to the cusps of G_{g}.
It turns out that lying behind monstrous moonshine is a certain string theory having the Monster group as symmetries; the conjectures made by Conway and Norton were proven by Richard Ewen Borcherds in 1992 using the noghost theorem from string theory and the theory of vertex operator algebras and generalized KacMoody superalgebras. Borcherds won the Fields medal for his work, and more connections between M and the jfunction were subsequently discovered.
Contents 
Formal versions of Conway's and Norton's conjectures
The first conjecture made by Conway and Norton was the socalled "moonshine conjecture"; it states that there is an infinitedimensional graded Mmodule
 <math>V = \bigoplus_{m\geq 1} V_m<math>
with <math>\dim(V_m) = c(m)<math> for all m, where
 <math>j({q}) = \sum_{m\geq 1} c_m {q}^m.<math>
From this it follows that every element g of M acts on each V_{m} and has character value
 <math>\chi_m(g) = \mathrm{tr}(g_{V_m})<math>
which can be used to construct the McKayThompson series of g:
 <math>T_g({q}) = \sum_{m\geq 1} \chi_m(g){q}^m.<math>
The second conjecture of Conway and Norton then states that with V as above, for every element g of M, there is a genus zero subgroup K of PSL_{2}(R), commensurable with the modular group Γ = PSL_{2}(Z), such that <math>T_g({q})<math> is the normalised main modular function for K.
The Monster module
It was subsequently shown by A. Oliver L. Atkin, Paul Fong and Frederic L. Smith using computer calculation that there is an indeed an infinitedimensional graded representation of the Monster group whose McKayThompson series are precisely the Hauptmoduls found by Conway and Norton, and I. B. Frenkel, J. Lepowsky and A. Meurman explicitly constructed this representation using vertex operators. The resulting module is called the Monster module.
Borcherds' proof
Richard Ewen Borcherds' proof of the conjecture of Conway and Norton can be broken into five major steps as follows:
 A vertex algebra V is constructed that is a graded algebra affording the moonshine representations on M, and it is verified that the monster module has a vertex algebra structure invariant under the action of M. V is thus called the Monster vertex algebra.
 A Lie algebra <math>\mathcal{M}<math> is constructed from V using the GoddardThorn "noghost" theorem from string theory; this is a generalized KacMoody Lie algebra.
 A denominator identity for <math>\mathcal{M}<math> is constructed that is related to the coefficients of <math>j({q})<math>.
 A number of twisted denominator identities are constructed that are similarly related to the series <math>T_g({q})<math>.
 The denominator identities are used to determine the numbers c_{m}, using Hecke operators, Lie algebra homology and Adams operations.
Thus, the proof is completed. Borcherds was later quoted as saying "I was over the moon when I proved the moonshine conjecture", and "I sometimes wonder if this is the feeling you get when you take certain drugs. I don't actually know, as I have not tested this theory of mine."
Why "monstrous moonshine"?
The term "monstrous moonshine" was coined by Conway, who, when told by John McKay in the late 1970s that the coefficient of <math>{q}<math> was precisely one more than the dimension of the Griess algebra (and thus exactly the degree of the smallest complex representation of the Monster group), replied that this was "moonshine" (a word that is used in English as a moniker for crazy or foolish ideas). Thus, the term not only refers to the Monster group M; it also refers to the perceived craziness of the intricate relationship between M and the theory of modular functions.
However, "moonshine" is also a slang word for illegally distilled whiskey, and in fact, the name may be explained in this light as well. The Monster group was investigated in the 1970s by mathematicians Fricke, Andrew Ogg and John G. Thompson; they studied the quotient of the hyperbolic plane by subgroups of SL_{2}(R), particularly, the normalizer Γ_{0}(p)^{+} of Γ_{0}(p) in SL(2,R). They found that the Riemann surface resulting from taking the quotient of the hyperbolic plane by Γ_{0}(p)^{+} has genus zero iff p is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71 (that is, a supersingular prime), and when Ogg heard about the Monster group later on and noticed that these were precisely the prime factors of the size of M, he wrote up a paper offering a bottle of Jack Daniel's whiskey to anyone who could explain this fact.
References
 John Horton Conway and Simon P. Norton, Monstrous Moonshine, Bull. London Math. Soc. 11, 308339, 1979.
 I. B. Frenkel, J. Lepowsky, and A. Meurman, Vertex Operator Algebras and the Monster, Pure and Applied Math., Vol. 134, Academic Press, 1988
 Richard Ewen Borcherds, Monstrous Moonshine and Monstrous Lie Superalgebras, Invent. Math. 109, 405444, 1992, online (http://math.berkeley.edu/~reb/papers/)
 Terry Gannon, Monstrous Moonshine: The first twentyfive years, 2004, online (http://arxiv.org/abs/math.QA/0402345)
 Terry Gannon, Monstrous Moonshine and the Classification of Conformal Field Theories, reprinted in Conformal Field Theory, New NonPerturbative Methods in String and Field Theory, (2000) Yavuz Nutku, Cihan Saclioglu, Teoman Turgut, eds. Perseus Publishing, Cambridge Mass. ISBN 0738202045 (Provides introductory reviews to applications in physics).
External links
 Moonshine bibliography (http://cicma.mathstat.concordia.ca/faculty/cummins/moonshine.refs.html)