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

Hi, overflowers.

I have a question concerning the torsion of elliptic curves over number fields.

Let us consider an elliptic curve $E$ defined over ${\mathbb Q}$. From the Weil pairing one can prove that, for all primes $p$, ${\mathbb Q}[\zeta_p] \subset {\mathbb Q}(E[p])$, where ${\mathbb Q}(E[p])$ is the algebraic extension generated by the coordinates of all points of order $p$ on $E$.

I was thinking whether this could also be true for all integers $n$, not necessarily prime. It is probably something well-known, but I can not find an appropriate reference, nor an easy counterexample.

Thanks a lot!!

share|cite|improve this question
You can use the Weil pairing to show that same for all integers $n$. This is corollary 8.1.1 in Silverman's The Arithmetic of Elliptic Curves, in the section titled The Weil Pairing. – Dror Speiser May 24 '13 at 8:34
Yes. Weil pairing applies to $n$-torsion too. – i707107 May 24 '13 at 8:35
Thanks, I knew I was missing something! – Chema Tornero May 24 '13 at 8:59

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.