For an elliptic curve $E/k$, let $\alpha$ be any endomorphism over $\bar{k}$ in End($E$) and let $[n]$ be the multiplication-by-n endomorphism. Show that for any n-torsion point $P \in E[n]$, we must have $\alpha(P) \in E[n]$.

I know and have just showed that $[n]$ is commutative with all endomorphisms but I'm not sure how to use that fact here. What's a good way to show $\alpha(P) \in E[n]$?

  • $\begingroup$ In general, if two endomorphims commute, each stabilizes the kernel of the other. $\endgroup$ – Captain Lama Apr 12 '16 at 16:45
  • $\begingroup$ @CaptainLama how exactly does that help here? $\endgroup$ – user3772119 Apr 12 '16 at 16:47
  • 1
    $\begingroup$ Well, you say you showed that $[n]$ commutes with all $\alpha$ and you want to prove that $\alpha$ stabilizes the kernel of $[n]$. $\endgroup$ – Captain Lama Apr 12 '16 at 16:48

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