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## 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 2009 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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This wikipedia article defines supersingular primes as those appearing in the prime factorization of the order of the Monster, although that's really not satisfying.

Another question: why is the sequence of exponents here almost strictly decreasing? (That is, why do small primes appear with larger exponents?)

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 In fact, I didn't read the WP before posting an answer but it's fun to see that my idea agrees with the collective brain of Wikipedia :) – Ilya Nikokoshev Oct 19 2009 at 18:20 Now, the question about larger exponents might benefit from more rephrazing. The way you ask, one could point out to a trivial fact that an average number has much larger 2-exponent then, say, 17-exponent. – Ilya Nikokoshev Oct 19 2009 at 18:22 Well, according to the same article, the fact that the supersingular primes (defined through a certain on modular curves) are precisely those diving the order of the Monster was one of the first observations related to "monstrous moonshine", and presumably the deep reason connecting the two is precisely the kind of thing that Borcherds' work elucidates... I'm certainly not qualified to review this connection, but at least it seems that this isn't a completely different question from the connection you pointed out between the dimensions of irreps of M and the j invariant. – Alon Amit Oct 19 2009 at 18:27 In fact, I copied it from my notes and yes, they appeared in my notes together because they are related! But may be there is a different answer? – Ilya Nikokoshev Oct 19 2009 at 19:06