MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
Never. An elliptic curve over a finite field always has an extra endomorphism, namely Frobenius. – user18237 Jul 14 '12 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. – 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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.