# Elliptic Curves over char $p$ with given endomorphism

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 $$Q_D(X)$$.