I encountered a theorem saying that, Given an elliptic curve $E_1$ and arbitrary subgroup $H$, there only exist an isogeny $\phi:E_1\rightarrow E_2$ with $\ker\phi=H$ up to isomorphism of its image. Here's my question, I want to prove this theorem, or at least get some intuition.
Just kindly remind that, an isogeny is a morphism on elliptic curves that preserve both group structure and projective variety. So simply giving a group homomorphism that eliminate $H$ would not suffice.
But since uniqueness can be given simply by group structure of $E_1$, the theorem is actually stating the existence of such algebraic morphism. In other word, the theorem is saying that the canonical morphism given by $x\mapsto x+H$, is algebraic, which means that its image must be also another elliptic curve. Any idea?