MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
Duplicate of…, 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

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.