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?