7
$\begingroup$

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.

$\endgroup$
3
  • $\begingroup$ Duplicate of mathoverflow.net/questions/55/supersingular-elliptic-curves/…, though I don't think anyone mentions the congruence subgroup POV. $\endgroup$ Oct 19, 2009 at 18:24
  • 1
    $\begingroup$ 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? $\endgroup$ Oct 19, 2009 at 18:30
  • $\begingroup$ Looking now at your moonshine question I realize I jumped the gun and that these are probably different notions. $\endgroup$ Oct 19, 2009 at 19:20

1 Answer 1

7
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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