TITLE: etale cohomology of an abelian variety and its dual
QUESTION [8 upvotes]: 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 $A^{*}$?

REPLY [8 votes]: Let me add something to what Timo Keller says rightly. What he said can be reformulated
as $H^i_{et}(A,{\bf Z_\ell})$ and $H^i_{et}(A^*,{\bf Z_\ell})(1)$ are canonically dual
of each other. This should be understand that there are dual not only as $\bf Z_\ell$-modules,
but also as $Gal(\bar k / k)$-representation, and here the (1) takes its significance: it is a twist by the $\ell$-adic cyclotomic character.
Now, more can be said. Every abelian variety $A$ over $k$ as a polarization $p$ over $k$, which is in particular a symmetric isogeny $p: A \rightarrow A^\ast$. Now any isogeny induces isomorphism between the $V_\ell$'s (that is $T_\ell \otimes \bf Q_\ell$), hence taking again
the wedge power, we get for all $i$ an isomorphism
$$H^i_{et}(A,{\bf Z_\ell}) \otimes {\bf Q_\ell} \rightarrow H^i_{et}(A^*,\bf Z_\ell) \otimes \bf Q_\ell.$$ This isomorphism preserves other structures (the $Gal(\bar k/k)$-action), but is not canonical, since it depends on the choice of $p$. But this is certainly a relation between the stale cohomology of $A$ and $A^\ast$ that is worth mentioning, and that is 
logically independent of the one given by Timo.