MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

## Reduction of endomorphism ring of Non-CM elliptic curve

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. $\overline{End(E)} \cong End(\overline{E})$?

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