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, $E\mathrm{/}k$ an elliptic curve, and $\mathrm{\ell }$ an integer coprime to $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}(k)$. Let $\mathfrak{h}$ be an ideal of $\mathrm{E}\mathrm{n}\mathrm{d}(E)$ dividing $\mathrm{\ell }$, and consider the corresponding torsion subgroup $E[\mathfrak{h}]\subseteq E[\mathrm{\ell }]$. 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 $\mathrm{G}\mathrm{a}\mathrm{l}(K\mathrm{/}k)$. This yields $(\mathrm{E}\mathrm{n}\mathrm{d}(E)\mathrm{/}\mathfrak{h}{)}^{\ast }\to \mathrm{G}\mathrm{a}\mathrm{l}(K\mathrm{/}k)$ a group homomorphism.