Given two elliptic curves $E$ and $E'$ such that $\phi:E\to E'$ is an isogeny with kernel $K\subset E(\mathbb{Q})$, what relation is there between $E(\mathbb{Q})$ and $E'(\mathbb{Q})$?
I know that when the elliptic curves are defined over an algebraically closed field, then isogenies are either surjective or constant (Hartshorne shows this). Using the dual isogeny, it is then possible to deduce some results about the relation between the groups from there. But now, working with $\mathbb{Q}$ is quite a bit different. Any ideas on how to establish useful morphisms between the groups of rational points of these two curves? My goal is to describe $E(\mathbb{Q})$ using $E'(\mathbb{Q})$ and $K$.
Thank you very much.
While $E'(\Bbb Q)$ contains a subgroup isomorphic to $E(\Bbb Q)/K$ that might not be all of it. But this subgroup does have finite index in $E'(\Bbb Q)$. This is because $E(\Bbb Q)$ and $E'(\Bbb Q)$ have the same rank (to prove this use the dual isogeny).
Playing off $E(\Bbb Q)$ against various $E'(\Bbb Q)$ is at the heart of descent theory of elliptic curves.