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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.
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    $\begingroup$ The Frobenius map can be $[n]$ if the elliptic curve is supersingular over a field of square order. $\endgroup$ Commented May 29, 2020 at 16:07
  • $\begingroup$ @AnginaSeng Thanks. So in what way is my analysis of why it cannot be incorrect? $\endgroup$
    – rogerl
    Commented May 29, 2020 at 16:21

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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. )

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  • $\begingroup$ Thanks, that explains how my initial comment is wrong. Do you have any thoughts on the numbered questions that I asked? $\endgroup$
    – rogerl
    Commented May 31, 2020 at 20:25
  • $\begingroup$ I don’t usually make analogies for this situation @rogerl , but let me think about your questions. $\endgroup$
    – Lubin
    Commented May 31, 2020 at 20:40

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