Automorphism groups of the fields of definition of torsion points of elliptic curves in characteristic ≥5

Abstract

For a field K of characteristic $p\ge 5$, let $E_{s,t}:y^2=x^3+sx+t$ be an elliptic curve defined over the function field $K(s,t)$ in two variables s and t. For a non-negative positive integer e and a positive integer N which is not divisible by p, we prove that if $K\supseteq F_p^{\mathrm{alg}}$, then the automorphism group of the normal extension $K(s,t)\left(E_{s,t}\left[p^{e}N\right]\right)$ over $K(s,t)$ is isomorphic to ${(Z/p^{e}Z)}^{\times }\times {\mathrm{SL}}_2(Z/NZ)$. Applying this result, we also determine the automorphism group of the normal extension $K(s,t)\left(E_{s,t}\left[p^{e}N\right]\right)$ for a general field K of characteristic $p\ge 5$.