Ways to characterize supersingular primes?

Question

I've read the definition, and it basically says p is a supersingular prime iff the fundamental domain of a group generated by \Gamma(p) and a matrix ((0, 1), (-p, 0)) is rational.

And there's a finite list of those, so it's kind of stupid to try to characterize a finite set.

But, anyway, what is the deep meaning of supersingular primes? What are different ways to characterize them?

This question actually arose when I posted a different question about one of the characterizations, the one related to Monster finite group. I hope to collect all possible answers from number theory here.

Duplicate of http://mathoverflow.net/questions/55/supersingular-elliptic-curves/399#399, though I don't think anyone mentions the congruence subgroup POV. — David Zureick-Brown, Oct 19 '09 at 18:24
Ok, I have to face the fact that I am very ignorant person. But how is the supersingular prime and a supersingular elliptic curve related? — Ilya Nikokoshev, Oct 19 '09 at 18:30
Looking now at your moonshine question I realize I jumped the gun and that these are probably different notions. — David Zureick-Brown, Oct 19 '09 at 19:20

score 7 · Answer 1 · answered Oct 20 '09 at 00:16

Supersingular primes are those primes p for which all supersingular elliptic curves over an algebraic closure of F_p have j-invariant in F_p. There is a theorem of Deuring that implies the j-invariant always lies in F_p² for any prime p, so supersingular primes form a rather distinguished class. From the standpoint of probabilistic heuristics, you should expect these primes to be rather small, since there are exactly (p-1)/24 supersingular elliptic curves over an algebraic closure of F_p, weighted by automorphisms.

Ways to characterize supersingular primes?

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