Let $A$ and $B$ be two abelian varieties over a field $k$ and let $l$ be a prime number not dividing the caracteristic of $k$. Let $\phi : A \to B \in Hom(A,B)$ be such that $\phi$ is zero on $A_{l^n}(\overline{k})$.
In arithmetic geometry by Cornell and Silverman, Milne says on page 124 in the proof of lemma 12.6 that $\phi$ is zero on $A_{l^n}$ because $A_{l^n}$ is an étale subgroup scheme of $A$. I don't understand why being étale implies this so I'd be grateful if someone could explain this
For a geometrically-reduced scheme $X$ of finite type over a field $k$ (e.g., an etale scheme over $k$), the $\bar{k}$ points are dense in $X_{\bar{k}}$.