Are **supersingular primes** and **supersingular elliptic curves** related?


(this was essentially a subquestion in [my earlier question](https://mathoverflow.net/questions/1249/ways-to-characterize-supersingular-primes), but still looks sufficiently different to me to deserve a separate post)



What does "supersingular" mean?
Let F be the finite field with p elements.


A supersingular elliptic curve is an elliptic curve E/F with the property that the endomorphism ring (ring of homomorphisms from E to E) of E over the algebraic closure of F\_p is has rank 4 as a Z-module. 


It is a theorem that End(E/F) has rank 2 or rank 4 (and that in the former case, End(E/F) is isomorphic to an order in an imaginary quadratic number ring whereas in the latter case, End(E/F) is isomorphic to an order in a quaternion algebra over Q). Hence, a supersingular elliptic curve over F\_p can be thought of as an elliptic curve over F with a "big" endomorphism ring.


It is a theorem that another characterization supersingular elliptic curves is that E/F is supersingular is if and only if the number of points on E/F is exactly p + 1 (edit: for p > 3, see Voloch's comment below).


I believe that the etymology of the term "supersingular" is as follows: if you start with an elliptic curve E/Q over Q, for all but finitely many primes p, reduction (mod p) gives an elliptic curve E/F. For a generic E/Q (specifically, one without "complex multiplication") then the set of primes such that reduction (mod p) turns E/Q into a supersingular E/F has asymptotic density 0. Such primes are called "supersingular primes for E/Q" - supersingular refers to "really unusual." The reductions for such primes are then called supersingular elliptic curves over F. I'm pretty sure that every elliptic curve over F is a (mod p) reduction of an elliptic curve over Q so that all supersingular elliptic curves arise in this way.


I'll remark (following Silverman) that a supersingular elliptic curve over F is not "singular" in the sense of algebraic geometry - by definition all elliptic curves are nonsingular.


I do not know much about supersingular primes in the context of monstrous moonshine. According to Wikipedia, a supersingular prime is a prime that divides the order of the monster group; and there are 15 such primes. Given E/Q, by a theorem of Elkies there will be infinitely many super singular primes p for E/Q. So it's difficult to imagine how the list of 15 supersingular (with respect to moonshine) primes could emerge from the notion of "supersingular elliptic curve." I imagine that the etymology of the term "supersingular" in the context of moonshine is again that that supersingular primes are special - but that they special in a completely different way from the supersingular primes for a elliptic curve over Q.


