2
$\begingroup$

Let $A$ be an abelian variety of dimension $g$ defined over a number field $K$. Suppose $A$ has a principal polarization and $\ell$ is a prime number. We have a Weil pairing:

$$ e_\ell: A[\ell]\times A[\ell] \rightarrow \mu_\ell $$

I dont know if the Weil pairing is unique, so my question is as follow: Given a subgroup $G\subset A[\ell]$ of order $\ell^2$, is there a Weil pairing $e_\ell$ (eventually depending on $G$) such that there exist $P,Q\in G$ and $e_\ell(P,Q)$ is a primitive $\ell$-th root of unity?

I know from here that for any point $P\in G$, there exists a point $Q\in A[\ell]$ such that $e_\ell(P,Q)$ is a primitive $\ell$-th root of unity. Thanks in advance for comments and answers.

$\endgroup$
1
  • 3
    $\begingroup$ I am not sure what you call a "Weil pairing on $G$", but if you mean the restriction of the Weil pairing (which is unique and canonical), the answer is obviously no: just take for $G$ a totally isotropic subspaces of $A[\ell]$. $\endgroup$
    – abx
    Nov 21, 2018 at 16:52

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.