This question arose while reading Noam Elkies’ paper ‘*The existence of infinitely many supersingular primes for every elliptic curve over $ \mathbb{Q}$ *‘.

We collect $ j$ -invariants of elliptic curves defined over $ \overline{\mathbb{Q}}$ such that they have complex multiplication by the ring $ \mathcal{O}_D = \frac{1}{2} \left( D+\sqrt{-D} \right)$ , where $ D$ is a positive integer congruent to $ 0$ or $ 3$ mod $ 4$ . These $ j$ -invariants are conjugate algebraic integers over $ \mathbb{Q}$ and we may construct the monic polynomial $ P_D(X)$ whose roots are precisely these $ j$ -invariants. Now we would like to reduce this polynomial mod $ p$ and study the roots of the reduced polynomial $ Q_D(X)$ .

Elkies’ characterises the roots of $ Q_D(X)$ as precisely those $ j$ -invariants that correspond to elliptic curves $ E_p$ over $ \overline{F_p}$ such that they posses an endomorphism $ \frac{1}{2} \left( D+\sqrt{-D} \right)$ (i.e. the ring $ \mathcal{O}_D$ may be embedded in $ End(E_p)$ ). He notes that the proof may be deduced using Deuring’s Lifting Lemma. Using this Lemma (the version Thm. 14, Page 184 of Lang’s *Elliptic Functions*) I see that if an elliptic curve over $ \overline{F_p}$ has such property we may assume it is obtained by reducing a CM curve over $ \overline{\mathbb{Q}}$ with endomorphism ring an order containing $ \mathcal{O}_D$ but I am unable to see why it’s $ j$ -invariant must satisfy the polynomial $ Q_D(X)$ .