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.