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:
- 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)?
- 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.
- 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.
Maybe a couple of examples, like fresh air, will clear the mind.
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.
( Thanks to @AnginaSeng for bringing these examples to mind. )