Let $q$ be a prime power and let $m$ be a positive integer coprime to $q$. Further, let $E_1(\mathbb F_q)$ and $E_2(\mathbb F_q)$ be two isogenous supersingular elliptic curves over a finite field $\mathbb F_q$ such that
$$E_i(\mathbb F_q)[m] \cong \mathbb Z/m\mathbb Z \times \mathbb Z/m\mathbb Z$$
for $i = 1, 2$. Suppose that an isogeny $\phi \colon E_1 \rightarrow E_2$ is separable and has degree coprime to $m$. So in particular $\phi$ is a one-to-one and onto map between $E_1[m]$ and $E_2[m]$.
Now suppose that you are given a pair of generators $P,Q$ of $E_1[m]$ and a pair of generators $R,S$ of $E_2[m]$. In this setting, I have three questions.
- For which matrices
$$ \left( \begin{matrix} a & b\\ c & d \end{matrix} \right) \in \operatorname{GL}_2(\mathbb Z) $$
does there exist an endomorphism $\sigma \colon E_1 \rightarrow E_1$ such that $\sigma(P) = aP+bQ$ and $\sigma(Q)=cP+dQ$?
How hard is it to answer the following question: does there exist an isogeny $\phi \colon E_1 \rightarrow E_2$ such that $\phi(P)=R$ and $\phi(Q) = S$? Of course in general it is difficult to compute $\phi$, but perhaps answering this question of existence is easier?
Given a matrix
$$ \left( \begin{matrix} a & b\\ c & d \end{matrix} \right) \in \operatorname{GL}_2(\mathbb Z) $$
and a fact that $\phi(P)=R$ and $\phi(Q)=S$ for some isogney $\phi$, does there exist an isogeny $\phi'$ such that $\phi'(P) = aR+bS$ and $\phi'(Q)=cR+dS$? For which matrices is this possible?