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The two examples I have in mind are curves and abelian varieties. To be precise, if $C$ is a smooth projective algebraic curve over a number field $K$, then, for a prime $l$

$H^1_{et}(C_{\bar{K}},\mathbb{Q}_l)\cong T_l(J(C))\otimes\mathbb{Q}_l,$

where $J(C)$ is the Jacobian of $C$. The isomorphism is as $G_K$-modules, where $G_K$ is the absolute Galois group of $K$. A similar statement is true for abelian varieties.

What about smooth projective varieties of dimension $d$? Is there some generalization of the Jacobian with Tate module dual to the first etale cohomology group? When applied to an abelian variety, does this construction return the original abelian variety?

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Yes, the generalized Jacobian is the Picard scheme and the result is basic Kummer theory, i.e. long cohomology exact sequences associated to the short exact sequences $0\to \mu_{\ell^n} \to \mathbb{G}_m \to \mathbb{G}_m \to 0$ of etale sheaves on $X$, together with the isomorphism $H^1_{et}(X, \mathbb{G}_m) = H^1(X, \mathcal{O}_X^\ast) = Pic(X)$. – Piotr Achinger Jul 3 '13 at 14:09
Ah, I should have been able to guess this might be the case. It would be very useful to me to know a reference covering the details, do you know one? – Tom163 Jul 4 '13 at 11:15

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