The automorphism group of the $p^{n}$-torsion points of an elliptic curve over a field of characteristic $p \ge 5$

TL;DR

This paper determines the automorphism group of the $p^n$-torsion points of a specific elliptic curve over a field of characteristic $p \,\ge 5$, showing it is isomorphic to the multiplicative group of units modulo $p^n$.

Contribution

It explicitly characterizes the automorphism group of the $p^n$-torsion extension for elliptic curves over fields with characteristic $p \,\ge 5$, revealing its structure as a multiplicative group.

Findings

Automorphism group is isomorphic to $(\mathbb{Z}/p^n\mathbb{Z})^{\times}$

Inseparable degree of the extension is $p^n$

Provides explicit description for the automorphism group structure

Abstract

For a field $K$ of characteristic $p\ge5$ and the elliptic curve $E_{s,t}: y^{2} = x^{3} + sx + t$ defined over the function field $K\left(s,t\right)$ of two variables $s$ and $t$, we prove that for a positive integer $n$, the automorphism group of the normal extension $K\left(s,t\right)\left(E_{s,t}\left[p^{n}\right]\right)/K\left(s,t\right)$ is isomorphic to $\left(\mathbb{Z}/p^{n}\mathbb{Z}\right)^{\times}$, and its inseparable degree is $p^{n}$.