Mumford's book on abelian varieties?
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Kevin BuzzardMay 15 '11 at 14:30

The $i$th etale cohomology group is the $i$th wedge power of the 1st, and the 1st is the dual of the Tate module, so if you understand the Tate module then you understand all the etale cohomology.
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Kevin BuzzardMay 15 '11 at 14:31

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Mumford's book on abelian varieties?
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Kevin BuzzardMay 15 '11 at 15:24

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Your question is a bit vague. For example, if you fix $A$ and let $q$ be large enough, the action is trivial. In general, since the action is continuous and $G_{\mathbf{F}_q}$ is pro-cyclic, you really only need to know the action of Frobenious $\Phi_q$. The characteristic polynomial of $\Phi_q$ is a monic polynomial of degree $2\dim(A)$ with integer coefficients, and its complex roots satisfy $|\alpha|=\sqrt{q}$. Beyond that and duality, the precise action depends on the particular abelian variety and field.
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Joe SilvermanMay 15 '11 at 16:55

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@unknown: your summary looks good to me (assuming $\ell$ doesn't divide $q$). The char poly of Frobenius has integer coefficients and is independent of $\ell$. That always struck me as amazing. Of course then the Weil conjectures are even more amazing.
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Kevin BuzzardMay 15 '11 at 20:55