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Let $E$ be an elliptic curve defined over a number field without complex multiplication and with ordinary reduction at a prime $p\in\mathbb{N}$. When is the reduction mod $p$ map a surjection on the endomorphism ring I.e. $\overline{End(E)} \cong End(\overline{E})$?

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    $\begingroup$ If the elliptic curve $E$ is defined over a number field, then you should change $p$ by a prime ideal $\wp$ with ordinary reduction. $\endgroup$ Commented Jul 27, 2012 at 14:23

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I believe the answer to your question is never. The best reference I could find is Lang's "Elliptic Functions'', Chapter 13.

Suppose that $E$ is an elliptic curve defined over a number field $L$ with good ordinary reduction at a prime $\wp$ of $L$, and assume further that $E$ does not have CM. Then, $\operatorname{End}(E)=\mathbb{Z}$. Now, the reduction of $E \bmod \wp$, denoted here by $\overline{E}$, is an elliptic curve defined over a field $k=\mathcal{O}_L/\wp$ of characteristic $p$, and since $E$ has good ordinary reduction, it follows from Theorem 5 (in Lang's Chapter 13, Section 2) that $\operatorname{End}(\overline{E})$ is necessarily an order $\mathcal{O}$ in a quadratic imaginary field $K$. Therefore, there cannot be a surjection from $\operatorname{End}(E)\cong \mathbb{Z}$ onto $\operatorname{End}(\overline{E})\cong \mathcal{O}$.

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  • $\begingroup$ Even without that theorem we know it is either an order in a quadratic imaginary field or an order in a quaternion algebra, so the same argument should work. $\endgroup$
    – Matt
    Commented Jul 27, 2012 at 15:55

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