For $m\ne2$ I want to show that if two automorphisms coincide on $E(m)$, which is the $m$-torsion subgroup of the elliptic curve $E$, then these automorphisms are the same.
The statement is very simple and intuitive, but I really can not make any real progress. I guess this should be related with the structure of E(m), which I do know, but I can not do any thing meaningful.