Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $K$ be a quadratic number field, and let $E_1$ and $E_2$ be two isogeneous elliptic curves over $K$. Assume we know that $j(E_1)^\sigma=j(E_2)$ where $\sigma$ is the generator of the Galois group of $K/Q$. Can we then say that some twist of $E_1$ is a $Q$-curve? If so, is there a good way of describing the necessary twist?

share|improve this question

1 Answer 1

up vote 2 down vote accepted

Yes -- a Q-curve is one whose geometric isogeny class is preserved by Galois, and that's evidently the case here. Of course there is no guarantee that E_1 and its Galois conjugate are isogenous over K. Is that the question you're asking? If so, I think there's a cohomological criterion for this due to Quer -- at least that's what I say in Remark 2.9 of my paper with Chris Skinner about this stuff:

http://www.math.wisc.edu/~ellenber/QcurveF.pdf

share|improve this answer
    
I was mostly wondering if there is a twist that makes E_1 a Q-curve over K which, as you point out, is not necessarily so, and the criteria for it is given by Quer. –  Soroosh Sep 2 '11 at 17:47

Your Answer

 
discard

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.