Ways to characterize supersingular primes? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T08:47:25Z http://mathoverflow.net/feeds/question/1249 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/1249/ways-to-characterize-supersingular-primes Ways to characterize supersingular primes? Ilya Nikokoshev 2009-10-19T18:10:12Z 2009-12-09T04:52:00Z <p>I've read the definition, and it basically says p is a <strong>supersingular prime</strong> iff the fundamental domain of a group generated by \Gamma(p) and a matrix ((0, 1), (-p, 0)) is rational.</p> <p>And there's a finite list of those, so it's kind of stupid to try to characterize a finite set.</p> <p>But, anyway, what is the <em>deep meaning</em> of supersingular primes? What are different ways to characterize them?</p> <p>This question actually arose when I posted a different question about one of the characterizations, <a href="http://mathoverflow.net/questions/1247/mystery-of-the-monster" rel="nofollow">the one related to Monster finite group</a>. I hope to collect all possible answers from number theory here.</p> http://mathoverflow.net/questions/1249/ways-to-characterize-supersingular-primes/1319#1319 Answer by S. Carnahan for Ways to characterize supersingular primes? S. Carnahan 2009-10-20T00:16:25Z 2009-10-20T00:16:25Z <p>Supersingular primes are those primes p for which all supersingular elliptic curves over an algebraic closure of F<sub>p</sub> have j-invariant in F<sub>p</sub>. There is a theorem of Deuring that implies the j-invariant always lies in F<sub>p<sup>2</sup></sub> 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 F<sub>p</sub>, weighted by automorphisms.</p>