# Weil pairings on abelian varieties restricted to subgroups of a given order

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.

• 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 $A[\ell]$. – abx Nov 21 '18 at 16:52