"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.