Automorphisms of finite fields from isogeny cycles

We develop an explicit geometric construction of automorphisms of finite fields arising from isogeny cycles. Let $k$ be a finite field, $E/k$ an elliptic curve, and $\ell$ an integer coprime to $\mathrm{char}(k)$. Let $\mathfrak{h}$ be an ideal of $\mathrm{End}(E)$ dividing $\ell$, and consider the corresponding torsion subgroup $E[\mathfrak{h}]\subseteq E[\ell]$. From the action of End(E) on $E[\mathfrak{h}]$, we construct the splitting field $K$ of the $x$-coordinates of points in $E[\mathfrak{h}]$ and the associated Galois group $\mathrm{Gal}(K/k)$. This yields $(\mathrm{End}(E)/\mathfrak{h})^* \to \mathrm{Gal}(K/k)$ a group homomorphism.