Let $f : E \to E'$ be an isogeny between elliptic curves (or abelian varieties), defined over a field $k$. Let $X$ be the kernel of $f$.
Let $L \supset k$ be an algebraically closed field.
Is it true that the natural map $X(\bar k) \to X(L)$ is a group isomorphism?
When $E' = E$ is any abelian variety and $f$ is the multiplication by an integer $m$ distinct from the characteristic of $k$, then $X(\bar k) \cong (\Bbb Z/m)^{2 dim(E)} \to X(L) \cong (\Bbb Z/m)^{2 dim(E)}$ is injective hence an isomorphism, because of cardinality. But what about more general cases?
Best Answer
By definition an isogeny has finite kernel, and the natural map is always injective, so I believe your argument would generalize.