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Let $X/k$ be a variety and $A/k$ be an abelian variety. Set $\mathcal{A} = A \times_k X$ and $\bar{X} = X \times_k \bar{k}$.

Is it true that $\mathcal{A}[\ell^n] = A[\ell^n] \times_k X$? And what is $\mathcal{A}[\ell^n](\bar{X})$?

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