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Suppose we have an abelian variety $A$ defined over the rational numbers $\mathbb{Q}$. It is known that the Weil pairing $$ e_\ell: A[\ell]\times A[\ell] \rightarrow \mu_\ell $$ is non degenerate, which means that for every non zero point $P\in A[\ell]$, there exists a non zero point $Q \in A[\ell] \setminus<P>$, such that $e_\ell(P,Q)$ is a primitive $\ell$ root of unity.

Is there such a variety with the property that for almost all primes $\ell$ and for each point $P\in A[\ell]$ we have that $e_\ell(P,Q)$ is a primitive $\ell$ root of unity for all $Q\in A[\ell] \setminus<P>$?

I am not expert in this topic, so any comment or suggestion is welcome.

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    $\begingroup$ You just have a vector space over $\mathbb{F}_{\ell}$ with a skew-symmetric form $\varphi $, and you want $\varphi (x,y)\neq 0$ as soon as $x$ and $y$ are not proportional. This is absurd (already for one fixed $\ell$). $\endgroup$
    – abx
    Commented Sep 18, 2019 at 15:59
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    $\begingroup$ Moreover the Weil pairing is between $A[\ell]$ and $\hat{A}[\ell]$, where $\hat{A}$ is the dual abelian variety. $\endgroup$
    – abx
    Commented Sep 18, 2019 at 16:04

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