I have a lot of questions about the following paragraph (reference), most of which are probably relatively easy to answer.
- What even is an $\mathbb F_q$-rational morphism? Is it a morphism over $\mathbb F_q$?
- If that's what it means, why is $\phi$ $\mathbb F_q$-rational iff $\pi(G) = G$?
- Why "Formally, [...]", what does that mean?
- $E[\ell] \cong (\mathbb Z / \ell \mathbb Z )^{\oplus 2}$ ($\ell$ is assumed to be prime). Why does $\pi$ act on $E[\ell]$ like an element of $GL_2(\mathbb F_\ell)$?
- Why is the order of $\pi$ in $GL_2(\mathbb F_\ell)$ the degree of the smallest extension of $\mathbb F_q$ where all $\ell$-isogenies of $E$ are defined?
Thanks for any hints.
(1) A morphism $\phi : \mathbb{P}^N \to \mathbb{P}^M$ is given by $M + 1$ homogeneous degree $d$ polynomials in $\bar{k}[x_0,...,x_{N}]$. If these polynomials are in $k[x_0,...,x_n]$ (possibly after scaling by $\lambda \in \bar{k}^*$) then $\phi$ is said to be $k$-rational. Equivalently if $\sigma(\phi(P)) = \phi(\sigma(P))$ for all $\sigma \in G_k$.
(2) An isogeny is determined by its kernel. It is a theorem that $\phi$ is definied over $k$ if and only if $Ker(\phi)$ is $G_k$-stable. Since $G_{\mathbb{F}_q}$ is topologically generated by the Frobenius this occurs if and only if $G = Ker(\phi)$ is stable under $\pi$.
(4) An element of $G_k$ acts as an automorphism of $E[l]$. Fixing a $\mathbb{F}_l$-basis for $E[l]$ the automorphisms are just elements of $GL_2(\mathbb{F}_l)$.
(5) Note $\pi^k$ acts trivially on $E[l]$ if and only $\pi^k$ stablises all cyclic subgroups of $E[l]$. By the same argument as (2) this occurs if and only if all isogenies are defined over the fixed field of (the closure of) ${\langle \pi^k \rangle} \subset G_k$. This fixed field is $\mathbb{F}_{q^k}$, which has degree $k$ over $\mathbb{F}_q$.