# Endomorphism rings of elliptic curves over finite fields

I understand that any elliptic curve $$E$$ defined over a finite field $$\mathbb{F}_q$$ has an endomorphism ring $$End_{\overline{\mathbb{F}}_q}(E)$$ that is strictly larger than $$\mathbb{Z}$$, since the Frobenius map $$x\mapsto x^q$$ is an endomorphism (which cannot be $$[n]$$ for any $$n$$ since it is the identity on $$\mathbb{F}_q$$ but not elsewhere). But after that, I'm somewhat confused conceptually:

1. I understand how to visualize complex multiplication for a curve defined over $$\mathbb{Q}$$: the curve arises from a lattice, and complex multiplication by $$z$$ is multiplication in $$\mathbb{C}$$ in the complex torus. Pushing this over to $$E$$ via $$\wp$$ results in essentially a rational function of points on the curve. Is there a more geometric way of visualizing endomorphisms of a curve defined over $$\mathbb{F}_q$$ as well (even in the case of an ordinary curve)?
2. Suppose $$E$$ defined over $$\mathbb{F}_q$$ is ordinary with endomorphism ring $$\mathcal{O}$$. Is there always some lift of $$E$$ to a complex elliptic curve with complex multiplication? Is there always a lift of $$E$$ to a curve whose endomorphism ring is $$\mathcal{O}$$? (I am familiar with Deuring's theorem which states that under certain conditions what I said above is true). Examples would be greatly appreciated.
3. An answer to #1 above may help me here, but I can't visualize how the Frobenius map acts as an element of a quadratic order in the ordinary case. Again an example would be very helpful.
• The Frobenius map can be $[n]$ if the elliptic curve is supersingular over a field of square order. Commented May 29, 2020 at 16:07
• @AnginaSeng Thanks. So in what way is my analysis of why it cannot be incorrect? Commented May 29, 2020 at 16:21

First let’s look at the $$2$$-supersingular curve $$E:Y^2+Y=X^3$$. You do the doubling and see that $$[2](\xi,\eta)=(\xi^4,\eta^4+1)$$. (Even more curiously, $$[4](\xi,\eta)=(\xi^{16},\eta^{16})$$ ). Thus $$E$$, as an $$\Bbb F_4$$-curve has $$\mathop{\mathbf f}_4=[-2]_E$$. Of course the above identities are quite independent of where $$\xi$$ and $$\eta$$ lie.
In the same way, $$Y^2=X^3-X$$, which is $$3$$-supersingular, has $$[-3](\xi,\eta)=(\xi^9,\eta^9)$$. Etc.