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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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Never. An elliptic curve over a finite field always has an extra endomorphism, namely Frobenius. –  user18237 Jul 14 '12 at 18:08
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Perhaps it is worth noting that Frobenius can be a rational integer (necessarily $\pm q^{n/2}$) but of course this forces the curve to be supersingular. –  user18237 Jul 14 '12 at 19:18
    
A reference for gb's comment: Lang's "Elliptic Functions", Chapter 13, Section 2, Theorem 5. –  Álvaro Lozano-Robledo Jul 27 '12 at 14:52

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