There are strong connections between the Hodge and the Tate conjectures, mainly at the level of similarities and analogies. To quote from an answer of Matthew Emerton on MathOverflow:
"[…] we also have a natural abelian (in fact Tannakian) category in play: in the complex case, the category of pure Hodge structures, and [when the field of definition $K$ is finitely generated over its prime subfield], the category of $\ell$-adic representations of $G_K$ (the absolute Galois group of $K$) (for some prime $\ell$, prime to the characteristic of $K$ in the case when $K$ is a finite field).
Now taking cohomology gives a functor from the category of smooth projective varieties to this latter category (via Hodge theory in the complex case, and the theory of étale cohomology in the other cases). The Hodge conjecture (in the complex case) and the Tate conjecture (in the other cases) then says that this functor is fully faithful.
Moreover, it is known that the Hodge conjecture for CM abelian varieties over $\mathbb{C}$ implies the Tate conjecture over finite fields (this was proven by J. Milne), and that the Tate conjecture for abelian varieties over finitely generated fields implies the Hodge conjecture over $\mathbb{C}$ (this was proven by P. Deligne I. Piatetski-Shapiro – see anon's answer) (see this workshop summary). However, I remember that during a seminar at my university, someone said that the Hodge conjecture is expected to be more difficult to solve than the Tate conjecture. Also, the Hodge conjecture (H), unlike the Tate conjecture (T), is part of the Millenium problems, which could suggest that H is more difficult/deeper that T. My question is thus the following: are there any reason why H should be more difficult than T?
Best Answer
Here is an argument that Tate is harder than Hodge:
Here is an argument that Hodge is harder than Tate:
The fact that both arguments seem reasonable shows (in my view) that the question doesn't have a good answer. Indeed, with our current state of knowledge being what it is (pretty poor for either conjecture), it is hard to compare the two --- they both seem like hard problems! They also seem to be closely coupled (and not just at the level of analogies).
One thing that the second point suggests is that it could be important to reduce the Hodge conjecture from the case of general complex varieties to the case of varieties over number fields (or, equivalently, varieties over $\overline{\mathbb Q}$). (I first learned this suggestion from Langlands.) That is probably also a very hard problem, but more accessible than the Hodge conjecture itself.