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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
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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

You have an exact sequence of sheaves for the étale topology:

$0\rightarrow \mu_{l^{n}} \rightarrow \mathbb{G}_m \rightarrow \mathbb{G}_m \rightarrow 0$.

If you take an injective resolution of this complex, you can see immediately that the p^power commute the diagramme in the first arrow of this complex.

By applying global section and the fact that we are in characteristic 0:

we have a short exact sequence of etale group cohomology: $0\rightarrow H^1_{et}(X_{\overline{K}},\mu_{l^{n}})\rightarrow H^{1}_{et}(X_{\overline{K}}, \mathbb{G}_m) \rightarrow H^{1}_{et}(X_{\overline{K}},\mathbb{G}_m) \rightarrow 0$. Wa have also a duality between $H^1_{et}(X_{\overline{K}},\mu_{l^{n}})$ and $H^1_{et}(X_{\overline{K}},\mathbb{Z}/l^{p^{n}})$, and it is easy to see using complex Cech that $H^{1}_{et}(X_{\overline{K}}, \mathbb{G}_m)$ is the picard group and $H^1_{et}(X_{\overline{K}},\mu_{l^{n}})$ classifies the invertible sheaf with trivialisation of its $l^{n}$ tensor power. Thus, by taking the projective limits over $n$ you find that $H^1_{et}(X_{\overline{K}},\mathbb{Z}_p)$ is the dual of the tate module of the Picard scheme of $X$.

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