For a field $\mathbb{ K}$ of characteristic $p\ge5$ containing $\mathbb{ F}_{ p}^{ \operatorname{ alg}}$ 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 $\left(\mathbb{ Z}/p^{ e}\mathbb{ Z}\right)^{ \times} \times \operatorname{ SL}_{ 2} \left(\mathbb{ Z}/N\mathbb{ Z}\right)$.
Categories: math.NT