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.

  • 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 '18 at 16:52

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