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There is a McKay-Thompson series for the Monster group, namely $T_{1A}$, responsible for,

$e^{\pi\sqrt{163}} \approx 640320^3 + 744$

Another one ($T_{2A}$) for,

$e^{\pi/2\sqrt{232}} \approx 396^4 -104$

And a third one ($T_{3A}$) for,

$e^{\pi/3\sqrt{267}} \approx 300^3 + 42$

It turns out, as proven by Conway, Norton, and Atkin, that this family of functions span a linear space of dimension 163. I found this so intriguing I had to write an article on it. See,

"The 163 Dimensions of the Moonshine Functions"

The Monster is the largest of the sporadic simple groups, and 163 is the largest d such that $Q(\sqrt{-d})$ has unique factorization. Do you think this is just a coincidence?

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The Monster is the largest of the sporadic simple groups. There are arbitrarily large simple groups (there are infinitely many primes, for example!) –  Mariano Suárez-Alvarez Jul 4 '11 at 21:38
    
Thanks. Has been fixed. –  Tito Piezas III Jul 4 '11 at 21:41
    
Also related is Euler's famous polynomial $x^2+x+41$, which is prime for $0\leq x\leq 39$, factors as $\left(x+\frac{1-\sqrt{-163}}{2}\right)\left(x+\frac{1+\sqrt{-163}}{2}\right)$. –  Eric Naslund Jul 4 '11 at 21:42
    
Yes, you plug $\tau = (1+\sqrt{-163})/2$ into the McKay-Thompson series T_1A with constant term 744, and you get -640320^3. Given the prime-generating polynomial P(n) = 2n^2+29 which is prime for n = {0 to 28} and plug the root of P(n) = 0 into T_2A, and you get 396^4. And so on. –  Tito Piezas III Jul 4 '11 at 21:47
    
It is worth reading the Wikipedia page on Heegner Numbers: en.wikipedia.org/wiki/Heegner_number There are a lot of interesting things there. (What I said above, and the relation between those prime generating polynomials and Heegner numbers are in the article.) –  Eric Naslund Jul 4 '11 at 21:48
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up vote 1 down vote accepted

Gukov-Vafa Comm Math Phys 246, pp 181-110 has a remark in section 5.4 page 198.

I am following this up. It seems a knowledge (at least for physicists) of Ishibashi states and the Narain momentum lattice may be helpful.

Work by Connes, Consani, Marcolli, Ramachandran relates quantum statistical mechanics and class field theory.

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