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This is probably a very easy question for those familiar with the arithmetic theory of abelian varieties because it is either possible or impossible for "trivial reasons".

Let $A$ be an abelian variety over a number field $K$, say principally polarized if needed later. I want to somehow describe what a full level $n$ structure ($n\geq 4$) on $A$ is in terms of the Galois representation(s) associated to $A$. Is this even possible?

To give a full level $n$ structure is by definition equivalent to giving an isomorphism of group schemes $$A[n] \to (\mathbf Z/n\mathbf Z)^{2g}.$$

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You already described it. The Galois action on $A[n]$ is trivial. If you want to see it $\ell$-adically, factor $n$. But if you are talking about seeing that the action on $A[n]$ is trivial by using an $\ell$ that doesn't divide $n$, I don't think that's possible. –  Felipe Voloch Jun 26 '13 at 19:34
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