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I know that given an elliptic curve $E$ we can define an addition $+$ over the set of points on the curve to make in an abelian group. However

Can we define multiplication on $E$ in a natural way so that $E$ can be made into a ring?

Edit:

Can we do this in a non-trivial way?

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  • $\begingroup$ Not in a useful way, no. Depending on your precise setting (ground field), sometimes not at all. $\endgroup$
    – user2035
    Commented Oct 11, 2011 at 20:17
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    $\begingroup$ This question is sufficiently vague as to have little content. First when you say $E$ is an elliptic curve and you want to make $E$ into a ring, you're presumably talking about the points of $E$. So you must specify a field. Second, if you simply want a ring "in a non-trivial way", the answer doesn't seem interesting. (Generally, if $G$ is an abstract abelian group, can one put a non-trivial ring structure on it?) Maybe the right question is: Can we find an algebraic map $E\times E\to E$ that we'll call multiplication and that, with +, satisfies the ring axioms. The answer to that is no. $\endgroup$
    – Joe Silverman
    Commented Oct 11, 2011 at 20:23
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    $\begingroup$ Joe -- take $G=\mathbb{Q}/\mathbb{Z}$. Since $G\otimes_{\mathbb{Z}}G=0$, there are no ring structures on $G$ at all. $\endgroup$
    – algori
    Commented Oct 11, 2011 at 22:32
  • $\begingroup$ .. apart from the zero one. $\endgroup$
    – algori
    Commented Oct 11, 2011 at 22:43

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