Since it is my first post, I want to say hello to everyone :)
I have a problem with exercise 17c from the book "Rational points on elliptic curves" by Silverman and Tate. I have given the elliptic curve $E: y^2=x^3+x$ and $K_n=\mathbb{Q}(i)(E[n])$ the field generated by $i$ and coordinates of points of order $n$ on $E$. The exercise asks to show that if for every element $s\in H=\textit{Gal}\big(K_n/\mathbb{Q}(i)\big)$ there is an integer $m$ such that
$$s^2(P)=mP$$
for all $P$ with order $n$, then $G=\textit{Gal}\big(K_n/\mathbb{Q}\big)$ is abelian.
My try: I tried to use representation of $G$ in $\text{GL}_2(\mathbb{Z}/n\mathbb{Z})$. $E$ has complex multiplication $\phi(x,y)=(-x,iy)$ and I can take a basis such that $\phi=\begin{pmatrix}
0 & * \\
1 & *
\end{pmatrix}.$
Because $\phi^2=-I$ so $\phi=\begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}.$
$H$ is abelian and every $s\in H$ commutes with $\phi$ so
$$s=\begin{pmatrix}
a & -c \\
c & a
\end{pmatrix}.$$
Because $s^2=mI$ so $2ac=0$. For every $t\in G-H$, $t\phi=-\phi t$ thus
$$t=\begin{pmatrix}
b & d \\
d & -b
\end{pmatrix}.$$
Now I know that I can assume $n=p^\alpha$ and if $p>2$ then $a=0$ or $c=0$. But when $a=0$ then $ts\neq st$ and I don't know what to do.
Thank you for your help