# Endomorphism rings of ordinary elliptic curves

Let's say $$p$$ is a prime and $$t\neq 0$$ is a trace of Frobenius that occurs over $$\mathbb{F}_p$$. The discriminant of the Frobenius polynomial is $$\Delta:=t^2-4p.$$ So we obtain $$4p=t^2-\Delta.$$ If $$E$$ is an elliptic curve over $$\mathbb{F}_p$$ with trace of Frobenius $$t$$, then the Frobenius, call it $$\sigma$$, generates an order in the endomorphism ring of $$E$$, $$End(E)$$. Symbollically, $$\mathbb{Z}[\sigma]\subseteq End(E).$$ Put another way, the discriminant of $$End(E)$$ divides $$\Delta$$.

Now $$\Delta$$ may not be the discriminant of a maximal order. We write $$\Delta=B\Delta_{\mathcal{O}_K}$$ where $$\Delta_{\mathcal{O}_K}$$ is the discriminant of the ring of integers of some imaginary quadratic field $$K$$ and $$B\geq 1$$. Suppose there is an integer $$b>1$$ with $$b^2\mid B$$ and $$b$$ prime to $$p$$. Then $$b$$, the conductor, is the index of the order of discriminant $$b\Delta_{\mathcal{O}_K}$$ in $$\mathcal{O}_K$$. There could be an elliptic curve over $$\mathbb{F}_p$$ whose endomorphism ring is the order of discriminant $$b\Delta_{\mathcal{O}_K}$$.

My question is, in general, does every possible order occur as an endomorphism ring of an elliptic curve over $$\mathbb{F}_p$$? Kohel seems to indicate this is the case in his thesis, but I can't find a proof. By work of Deuring and because the $$j$$-invariants in question are algebraic, we know that over some finite extension of $$\mathbb{F}_p$$ there is an elliptic curve with an endomorphism ring that is isomorphic to each possible order. For my applications though, I'd like to have this elliptic curve be defined over $$\mathbb{F}_p.$$