Please apologize, if that's a stupid question. Why is the Frobenius Endomorphism of an elliptic curve over a finite field not regarded as an automorphism?
- Since it is an Isogeny, it is surjective
- Since it has trivial kernel, it is injective
Why is separability required?
Because it doesn't induce an automorphism of the function field of the curve. Automorphisms of an elliptic curve $E$ are endomorphisms $\varphi \colon E\to E$ which have degree $1$. Recall that the degree is defined as $[k(E)\colon \varphi^*(k(E)) ]$, where $k(E)$ is the function field of the curve and $\varphi^*$ is the induced map on $k(E)$. The Frobenius has degree $p$, hence it is not an automorphism. This somehow reflects the fact that map of curves are NOT maps of the underlying points, but maps of the underlying points AND of function fields.