"The Hodge conjecture says that a rational cohomology class on a nonsingular projective variety over C is algebraic if it is of type (p,p). The Tate conjecture says that a l-adic cohomology class on a nonsingular projective variety over a finitely generated field k is in the span of the algebraic classes if it is fixed by the Galois group. (A field is finitely generated if it is finitely generated as a field over its prime field.)" The Work of Tate by Milne.
You can see the analogy of Tate and Hodge conjectures.
"Pohlmann (1968) proved that the Hodge and Tate conjectures are equivalent for CM abelian varieties, Piatetski-Shapiro (1971) proved that the Tate conjecture for abelian varieties in characteristic zero implies the Hodge conjecture for abelian varieties,and Milne (1999) proved that the Hodge conjecture for CM abelian varieties implies the Tate conjecture for abelian varieties over finite fields." The work of Tate by Milne