The Tate conjecture: Let $k$ be a finitely generated field, $X/k$ a smooth projective geometrically integral variety and $\ell$ invertible in $k$. Then the cycle class map $$\mathrm{CH}^r(X) \otimes_\mathbf{Z} \mathbf{Q}_\ell \to \mathrm{H}^{2r}(\bar{X},\mathbf{Q}_\ell(r))^{G_k}$$ is surjective.

It is e.g. proved for $r=1$ and Abelian varieties, a deep theorem. See http://www.math.harvard.edu/~chaoli/doc/TateConjecture.html.

This is analogous to the Hodge conjecture for complex varieties.