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

Abstract

For a field $K$ of characteristic $p\ge 5$ 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 $p^n$.

Replacement of reference [8] in p.1 thanks to Davide Lombardo. 16pages, 1 figure