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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.

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  • $\begingroup$ Duplicate of mathoverflow.net/questions/55/supersingular-elliptic-curves/…, though I don't think anyone mentions the congruence subgroup POV. $\endgroup$
    – David Zureick-Brown
    Commented Oct 19, 2009 at 18:24
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    $\begingroup$ 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? $\endgroup$
    – Ilya Nikokoshev
    Commented 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. $\endgroup$
    – David Zureick-Brown
    Commented Oct 19, 2009 at 19:20

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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.

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