Automorphisms of finite fields from isogeny cycles

TL;DR

This paper constructs explicit automorphisms of finite fields using isogeny cycles on elliptic curves, linking endomorphism actions to Galois groups of splitting fields of torsion points.

Contribution

It provides a geometric method to realize automorphisms of finite fields via isogeny cycles and endomorphism actions on elliptic curve torsion points.

Findings

Explicit construction of automorphisms from isogeny cycles

Connection between endomorphism groups and Galois groups

Development of a homomorphism from endomorphism quotients to Galois groups

Abstract

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.