# Mystery of the Monstrous Moonshine

There's a very famous group, the largest sporadic simple finite group, sometimes called a monster whose size is quoted below. What's the explanation that the primes appearing in it,

#{Monster} = 2^{46} * 3^{20} * 5^9 * 7^6 * 11^2 * 13^3 * 19 * 23 * 29 * 31 * 41 * 47 * 59 * 71

are exactly the supersingular primes (and here's a separate question about those)?

My notes contain some mystic reference that it's "related to famous modular function $j(\tau) = q^{-1} + 744 + 196884q + \cdots$ by some compactification of bosonic strings on a Leech lattice". But perhaps there could be a more purely number-theoretic direction?

Also, here's a Wikipedia article with some references.

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Typo corrected, thanks. –  Ilya Nikokoshev Oct 19 '09 at 18:22

I can give you half of the answer, but the other half is wide open. I will use the characterization of supersingular primes as those primes p for which the normalizer of Gamma0(p) in SL(2,R) acts on the complex upper half plane to yield a genus zero quotient.

The monstrous moonshine conjecture asserted the existence of an infinite dimensional graded representation of the monster satisfying some exceptional properties. It was conjectured by Conway and Norton, and proved by Borcherds, using the representation constructed by I. Frenkel, Lepowsky, and Meurman. One can take the graded dimension of this representation to get a power series, and it is the q-expansion of the J-function. Furthermore, the graded trace of any element of order n in the monster is the q-expansion of a genus zero modular function that is invariant under Gamma0(nh) for some h|(12,n). One can conclude somewhat abstractly that the normalizer of Gamma0(p) in SL(2,R) has to be genus zero for any prime p dividing the order of the monster.

The proof I've seen that no other primes satisfy the genus zero condition does not seem to have anything to do with the monster. Instead, it is a delicate construction by Mazur involving the Eisenstein ideal, combined with some computations by H. Lenstra. I may be ignorant of more refined arguments developed in the last 30 years, though. [Edit: FC has pointed out that the proof of the bijection is a reasonably straightforward calculation. Still, I haven't seen any good arguments explaining the universality of the monster with respect to the genus zero property.]

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Scott has given us the group-theoretic connection involving GZ(p)+. There is a number- theoretic connection too.

Are not the monstrous primes just those for which all elliptic curves in char p have no p-torsion?

See also the Math Review of the papeer by Cord Erdenberger .. On the Kodaira dimension...

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Sorry, this isn't an answer but just a piece of related information for people who might be in the Cambridge (UK) area:

Marcus du Sautoy (Simonyi Professor for the Public Understanding of Science) is giving a public talk tonight entitled "Monstrous Moonshine". So if you want to see how someone might go about explaining the monster group to the general public then it might be a worthy talk. Perhaps I could ask Ilya's question at the end. :)

I think that it's worthwhile going just for some inspiration on how to present maths. He's a good communicator.

Details on talks.cam.ac.uk

(on another note, if anyone with special powers could fix my account I'd be very grateful, this could have been a comment!)

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