nt.number theory - A question on liftings of supersingular elliptic curves over the prime fields

I don't know a result that says what you want, but I'll give you some pointers that might help. The "number" of elliptic curves over $mathbb{F}_p$ with $p-t+1$ points is the "number" of binary quadratic forms with discriminant $H(t^2-4p)$. I put number in quotes because, on both sides of the formula, one must weight the objects being counted by one over the size of their automorphism group. (This will usually be $2$, in both cases.) I got this from section 1 of Lenstra's paper Factoring Integers with Elliptic Curves, which is very readable, he says that the result is essentially due to Deuring.

In chapter 13, section 5, of Lang's Elliptic Functions, I find the following statement, again attributed to Deuring: Given an elliptic curve $E_0$ over $mathbb{F}_p$, and an endomorphism $a_0$ of $E_0$, there is a number field $K$, an elliptic curve $E$ over $K$, an endomorphism $a$ of $E$ and a place $mathfrak{p}$ of $mathcal{O}_K$, lying over $p$, such that the reduction of $(E, a)$ modulo $mathfrak{p}$ is the base change of $(E_0, a_0)$ to $mathcal{O}_K/mathfrak{p}$. (Lang doesn't mention this base change explicitly, so maybe I am missing something; it seems to me to be what he is proving.) So, in particular, if $E_0$ has $p-t+1$ points, it can be lifted to a curve with complex multiplication by $mathbb{Z}[F]/(F^2-tF+p)$.

What Lang doesn't say, though, is anything about the uniqueness of this lift. And, indeed, there are some issues here, because any quadratic twist of $E$, by a $D$ for which $left( frac{D}{p} right) =1$, would have the same properties.

• Next nt.number theory - Where stands functoriality in 2009?
• Previous Do other stars have Oort clouds?