From Weil conjecture we know the relation between the zeta-function and the cohomology of the variety, however it appears that there are certainly more information containing in the zeta-function, and the question remains whether they can be used to compute some more geometric invariants of the variety, such as the Chern classes. For instance, can one spot a Calabi-Yau manifold just by looking at the zeta-function? Is the zeta-function a birational invariant, or stronger?
And consider the roots and poles of the zeta-function. Their absolute values are determined by the Riemann Hypothesis, nevertheless the "phases" still appear to be very mysterious. Are that any good explanations for them, e.g. in the case of elliptic curves?
Best Answer
Although the question is phrased a bit sloppily, there is a standard interpretation: Given a smooth complex proper variety $X$, choose a smooth proper model over a finitely generated ring $R$. Then one can reduce modulo maximal ideals of $R$ to get a variety $X_m$ over a finite field, and ask what information about $X$ it retains. As has been remarked, the zeta function of $X_m$ gives back the Betti numbers of $X$.
I believe Batyrev shows in this paper that the zeta function of a Calabi-Yau is a birational invariant and deduces from this the birational invariance of Betti numbers for Calabi-Yau's. And then, Tetsushi Ito showed here that knowledge of the zeta function at all but finitely many primes contains info about the Hodge numbers. (He did this for smooth proper varieties over a number field, but a formulation in the 'general' situation should be possible.)
For an algebraic surface, once you have the Hodge numbers, you can get the Chern numbers back by combining the fact that
$c_2=\chi_{top},$
the topological Euler characteristic, and Noether's formula:
$\chi(O_X)=(c_1^2+c_2)/12.$
I guess this formula also shows that if you know a priori that $m$ is a maximal ideal of ordinary reduction for both $H^1$ and $H^2$ of the surface, then you can recover the Chern numbers from the zeta functions, since $H^1(O_X)$ and $H^2(O_X)$ can then be read off from the number of Frobenius eigenvalues of slope 0 and of weights one and two.
You might be amused to know that the homeomorphism class of a simply-connected smooth projective surface can be recovered from the isomorphism class of $X_m$. (One needs to formulate this statement also a bit more carefully, but in an obvious way.) However, not from the zeta function. If you compare $P^1\times P^1$ and $P^2$ blown up at one point, the zeta functions are the same but even the rational homotopy types are different, as can be seen from the cup product in rational cohomology. See this paper.
Added: Although people can see from the paper, I should have mentioned that Ito even deduces the birational invariance of the Hodge (and hence Betti) numbers for smooth minimal projective varieties, that is, varieties whose canonical classes are nef. Regarding the last example, I might also point out that this is a situation where the real homotopy types are the same.
Added again: I'm sorry to return repeatedly to this question, but someone reminded me that Ito in fact does not need the zeta function at 'all but finitely many primes.' He only needs, in fact, the number of points in the residue field itself, not in any extension.