The automorphism group of torsion points of an elliptic curve over a field of characteristic $\ge 5$

Abstract

For a field $\mathbb{K}$ of characteristic $p\ge 5$ containing ${\mathbb{F}}_p^{\mathrm{alg}}$ and the elliptic curve $E_{s,t}:y^2=x^3+sx+t$ defined over the function field $\mathbb{K}\left(s,t\right)$ of two variables $s$ and $t$, we prove that for a non-negative positive integer $e$ and a positive integer $N$ which is not divisible by $p$, the automorphism group of the normal extension $\mathbb{K}\left(s,t\right)\left(E_{s,t}\left[p^{e}N\right]\right)$ over $\mathbb{K}\left(s,t\right)$ is isomorphic to ${\left(\mathbb{Z}/p^e\mathbb{Z}\right)}^{\times }\times {\mathrm{SL}}_2\left(\mathbb{Z}/N\mathbb{Z}\right)$.

16 pages, 1 figures