This is from Silverman's Arithemetic of Elliptic Curves, Chapter 8, section 1 :
I have been able to undertsand why the left kernel is $mE(K)$ and the right kernel is $G_{\bar{K}/L}$ but what I don't understand is the statement, 'Hence, ...'
I think by part (c), the map $E(K)/mE(K) \times G_{\bar{K}/K} \mapsto E[m]$ is injective on the left, and similar to part (d), induces a map $ G_{\bar{K}/K} \to Hom(E(K)/mE(K), E[m])$, with kernel $G_{\bar{K}/L}$, and hence induces an injective map $ G_{L/K} \to Hom(E(K)/mE(K), E[m])$.
Similarly, we have the map $ E(K)/mE(K) \to Hom(G_{L/K}, E[m])$ injective.
But don't we also need to show that these two maps are surjective to show the 'perfect' claim in the statement?
It would be great if someone could comment on my above reasoning and maybe clarify what the author means by 'perfect' here?
Thank you very much.