TITLE: Ways to characterize supersingular primes?
QUESTION [7 upvotes]: 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.

REPLY [7 votes]: Supersingular primes are those primes p for which all supersingular elliptic curves over an algebraic closure of Fp have j-invariant in Fp.  There is a theorem of Deuring that implies the j-invariant always lies in Fp2 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 Fp, weighted by automorphisms.