Trace of elliptic curve in CM method

Question

I am trying to understand the CM method for elliptic curves. Suppose we fix a discriminant $D<0$ and a prime $p$. In the CM method, we look for integer solutions $(t,y)$ to the norm equation $4p = t^2 -Dy^2$. If these solutions exist, then we can construct an elliptic curve over $\mathbb{F}_p$ with $p+1 \pm t$ rational points. What I find it hard to understand is that this step only involves studying the Hilbert polynomial $H_D$ modulo $p$, and we recover a curve with trace $\pm t$ (and not some other $t'$ that might satisfy the norm equation).

My question is, is the solution $(t,y)$ to the norm equation $4p=t^2-Dy^2$ unique if we fix $D$ and $p$ (and assume $t,y>0$)? If so, why?

Answer 1

The solution to the norm equation is unique, under the additional assumptions we make on $D$. Recall that we require either that $D \equiv 1 \mod 4$ or that $D \equiv 0 \mod 4$.

Moreover, your claim about recovering such a curve is only true, if in addition, $D \notin \{-3,-4\}$.

The reason for this uniqueness result is the fact that the group of units in $\mathbb{Z}\left[ \frac{D + \sqrt{D}}{2} \right]$ is $\{\pm 1\}$.

(For reference, see any standard textbook on algebraic number theory)

It follows that if two elements have the same norm, their ratio must have norm 1, hence is a unit, hence in these cases they differ only by a sign.