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Question 1:

Let $E_1,E_2$ be two elliptic curves over a field $k$ of characteristic zero with complex multiplication. Let $R_1,R_2$ be the endomorphism rings of the elliptic curves and suppose that the fraction fields of $R_1,R_2$ are equal. Is it necessarily the case that $E_1,E_2$ are isogenous?

Question 2:

Let $E$ be an elliptic curve over a field $k$ of characteristic $0$ with complex multiplication by $R$. Let $K$ be the fraction field of $R$ and let $S$ be the ring of integers in $R$. Can we construct an elliptic curve $E'$ with complex multiplication by $S$ which is isogenic to $E$?

Can we do this over $k$?

This is all clear if $k = \mathbb C$ but what about the general case? I am most interested in $k = \mathbb Q$. Perhaps you can do a Lefschetz style proof but I would rather not.

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  • $\begingroup$ Are you assuming the curves have CM? $\endgroup$ – Lord Shark the Unknown Jul 19 '17 at 17:31
  • $\begingroup$ Yes, I should have made this clear. $\endgroup$ – Asvin Jul 19 '17 at 17:40
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    $\begingroup$ what about $y^2 = x^3+1$ and $y^2 = x^3+2$ ? $\endgroup$ – mercio Jul 19 '17 at 20:50
  • $\begingroup$ I am confused. I think they have endomorphisms by orders in $\Bbb Z[\omega]$ but sage tells me there is no isogeny between them. But surely I can think of them as complex curves and construct an isogeny by hand? $\endgroup$ – Asvin Jul 19 '17 at 21:19
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    $\begingroup$ Ah, you mean the isogeny is not defined over Q. $\endgroup$ – Asvin Jul 20 '17 at 0:35

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