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Let $E_1$ be an elliptic curve over $\Bbb Q$ and $S \subset E_1(\Bbb Q)$ be a subgroup.

Is there an elliptic curve $E_2$ with an algebraic group morphism $E_1 \to E_2$, and such that $E_2(\Bbb Q) \cong E_1(\Bbb Q) / S$ ? Or such that $E_2(\overline{\Bbb Q}) \cong E_1(\overline{\Bbb Q}) / S$ (seeing $E_1(\Bbb Q)$ as a subgroup $E_1(\overline{\Bbb Q})$)?

According to Silverman's books on elliptic curves, this holds if $S$ is finite, but what if $S$ has rank $1$, for instance?

Thank you!

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    $\begingroup$ A surjective algebraic map between elliptic curves will have a finite kernel. $\endgroup$ – Lord Shark the Unknown Aug 26 '18 at 13:06
  • $\begingroup$ @LordSharktheUnknown: ok, but actually I don't need surjectivity for the map, I only need to have the $\Bbb Q$-points (or the $\overline{\Bbb Q}$-points) of $E_2$ to be a quotient of those of $E_1$. What can we say then ? $\endgroup$ – Alphonse Aug 26 '18 at 13:12

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