Let $A$ an abelian variety over a field $k$ and $A^{\*}$ the dual abelian variety.


How can we relate the étale cohomology of $A$ with etale cohomology of $A^{\*}$?



etale cohomology of an abelian variety and its dual
The Weil pairing induces a perfect pairing $T\_\ell A \times T\_\ell A^\vee \to \mathbf{Z}\_\ell(1)$, and $T\_\ell A = H^1\_{et}(\bar{A},\mathbf{Z}\_\ell)^\vee$ (here, $\bar{A}$ denotes $A \times\_k \bar{k}$). Further, $H^i\_{et}(\bar{A},\mathbf{Z}\_\ell) = \wedge^iH^1\_{et}(\bar{A},\mathbf{Z}\_\ell)$. One also has $A^\vee = \mathcal{E}\mathrm{xt}\_k^1(A,\mathbf{G}\_m)$.


