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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7$\begingroup$ Never. An elliptic curve over a finite field always has an extra endomorphism, namely Frobenius. $\endgroup$– user18237Commented Jul 14, 2012 at 18:08
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5$\begingroup$ 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. $\endgroup$– user18237Commented Jul 14, 2012 at 19:18
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$\begingroup$ A reference for gb's comment: Lang's "Elliptic Functions", Chapter 13, Section 2, Theorem 5. $\endgroup$– Álvaro Lozano-RobledoCommented Jul 27, 2012 at 14:52
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