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Can someone give me references for the structure of the $G_{\mathbf{F}_q}$-module $T_\ell A$, $A/\mathbf{F}_q$ an abelian variety?

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Mumford's book on abelian varieties? –  Kevin Buzzard May 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. –  Kevin Buzzard May 15 '11 at 14:31
Mumford's book on abelian varieties? –  Kevin Buzzard May 15 '11 at 15:24
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. –  Joe Silverman May 15 '11 at 16:55
@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. –  Kevin Buzzard May 15 '11 at 20:55

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