Infinite subgroups of elliptic curves and quotients

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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
$E_2(overlineBbb Q) cong E_1(overlineBbb Q) / S$ (seeing $E_1(Bbb Q)$ as a subgroup $E_1(overlineBbb Q)$)?



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!




elliptic-curves arithmetic-geometry




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  • 2




    A surjective algebraic map between elliptic curves will have a finite kernel.
    – Lord Shark the Unknown
    Aug 26 at 13:06










  • @LordSharktheUnknown: ok, but actually I don't need surjectivity for the map, I only need to have the $Bbb Q$-points (or the $overlineBbb Q$-points) of $E_2$ to be a quotient of those of $E_1$. What can we say then ?
    – Alphonse
    Aug 26 at 13:12














up vote
0
down vote

favorite












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
$E_2(overlineBbb Q) cong E_1(overlineBbb Q) / S$ (seeing $E_1(Bbb Q)$ as a subgroup $E_1(overlineBbb Q)$)?



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!




elliptic-curves arithmetic-geometry




share|cite|improve this question


















  • 2




    A surjective algebraic map between elliptic curves will have a finite kernel.
    – Lord Shark the Unknown
    Aug 26 at 13:06










  • @LordSharktheUnknown: ok, but actually I don't need surjectivity for the map, I only need to have the $Bbb Q$-points (or the $overlineBbb Q$-points) of $E_2$ to be a quotient of those of $E_1$. What can we say then ?
    – Alphonse
    Aug 26 at 13:12












up vote
0
down vote

favorite









up vote
0
down vote

favorite











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
$E_2(overlineBbb Q) cong E_1(overlineBbb Q) / S$ (seeing $E_1(Bbb Q)$ as a subgroup $E_1(overlineBbb Q)$)?



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!




elliptic-curves arithmetic-geometry




share|cite|improve this question














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
$E_2(overlineBbb Q) cong E_1(overlineBbb Q) / S$ (seeing $E_1(Bbb Q)$ as a subgroup $E_1(overlineBbb Q)$)?



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!





elliptic-curves arithmetic-geometry





share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 26 at 13:11

























asked Aug 26 at 13:02









Alphonse

1,815622




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  • 2




    A surjective algebraic map between elliptic curves will have a finite kernel.
    – Lord Shark the Unknown
    Aug 26 at 13:06










  • @LordSharktheUnknown: ok, but actually I don't need surjectivity for the map, I only need to have the $Bbb Q$-points (or the $overlineBbb Q$-points) of $E_2$ to be a quotient of those of $E_1$. What can we say then ?
    – Alphonse
    Aug 26 at 13:12












  • 2




    A surjective algebraic map between elliptic curves will have a finite kernel.
    – Lord Shark the Unknown
    Aug 26 at 13:06










  • @LordSharktheUnknown: ok, but actually I don't need surjectivity for the map, I only need to have the $Bbb Q$-points (or the $overlineBbb Q$-points) of $E_2$ to be a quotient of those of $E_1$. What can we say then ?
    – Alphonse
    Aug 26 at 13:12







2




2




A surjective algebraic map between elliptic curves will have a finite kernel.
– Lord Shark the Unknown
Aug 26 at 13:06




A surjective algebraic map between elliptic curves will have a finite kernel.
– Lord Shark the Unknown
Aug 26 at 13:06












@LordSharktheUnknown: ok, but actually I don't need surjectivity for the map, I only need to have the $Bbb Q$-points (or the $overlineBbb Q$-points) of $E_2$ to be a quotient of those of $E_1$. What can we say then ?
– Alphonse
Aug 26 at 13:12




@LordSharktheUnknown: ok, but actually I don't need surjectivity for the map, I only need to have the $Bbb Q$-points (or the $overlineBbb Q$-points) of $E_2$ to be a quotient of those of $E_1$. What can we say then ?
– Alphonse
Aug 26 at 13:12















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