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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?

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I am bewildered by this question. The title is "how much complex geometry does..." but then you talk about the Weil conjectures, which are about algebraic varieties over finite fields. Your question is about varieties---but over which base field? –  Kevin Buzzard Mar 30 '10 at 9:36
    
Hi Kevin, what i mean is whether it is possible to spot the complex geometry of the variety over C, from the zeta-function which is about the variety over F_{p^n}. And if the local zeta-function is not enough, will the global L-function be so? –  Bo Peng Mar 30 '10 at 9:45
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I think the question is something like this: if we know the zeta function over F_p, then we know the Betti numbers over C. But the zeta function has more information than the Betti numbers: it also has Frobenius eigenvalues, and maybe they give us additional information about the complex variety. –  Qiaochu Yuan Mar 30 '10 at 11:50
    
Doesn't the action of Aut(C) preserve the zeta-function but generally change the complex geometry? If I'm right, such examples would put limits on what the zeta function can know. –  David Feldman Dec 20 '10 at 17:21

3 Answers 3

up vote 17 down vote accepted

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.

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+1: a very useful answer. To go along with my previous comment, let me note that Batyrev construes Calabi-Yau in the weakest possible way (and thus has the strongest possible result): he requires only that the canonical bundle be trivial. So in particular this includes abelian varieties. –  Pete L. Clark Mar 30 '10 at 23:06
    
Very nice answer! But either you have a minor error at the end or I misunderstand you. $P^1 \times P^1$ is not homotopic to $P^2$ blown up at a point. In $H^2(P^1 \times P^1, \mathbb{Z})$, every class has even self intersection; this is not true in $P^2$ blown up at a point. –  David Speyer Mar 31 '10 at 12:16
    
Hmm. Perhaps I'm mistaken, but the real homotopy type is determined by the real cohomology ring, according to the theorem of Deligne-Griffiths-Morgan-Sullivan. This is not saying the spaces are homotopic. As mentioned, they aren't even rationally homotopic, exactly because of the intersection form you allude to. –  Minhyong Kim Mar 31 '10 at 12:21
    
I see, I did not realize that "real homotopy" was a technical term. Thanks! –  David Speyer Mar 31 '10 at 13:50

For an elliptic curve the global zeta function should determine it up to isogeny. You cannot expect anything more. I am not much of an analytic number theorist but I believe you might be able to extract the number field if you are given an analytic function which is the L-function of some elliptic curve over some number field. Once this is done, you can read off the traces of Frobenius, hence the $\ell$-adic rep, hence the curve up to isogeny.

In general, I think the question is too illl-posed to get meaningful answers.

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This is essentially a string of comments that would not fit into the box, with one answer in the middle.

@Kevin: for instance, we could take $X$ to be a variety over a $p$-adic field $K$ with good reduction. I agree that the question should begin by making this explicit. The Betti numbers have an interpretation in terms of $\ell$-adic cohomology, so are independent of the choice of embedding $K \hookrightarrow \mathbb{C}$.

@OP: Note that the zeta function cannot be a birational invariant because the Betti numbers are not birational invariants: if you blow up e.g. a surface at a point, $B_2$ increases by $1$. The fact that, as Felipe says, the zeta function is a rational isogeny invariant of abelian varieties -- along with the implication that, for algebraic curves, the zeta function depends only on the isogeny class of the Jacobian -- is the strongest invariance statement I know of along these lines.

The question about Calabi-Yau's seems interesting, although one should say exactly what one means by a Calabi-Yau variety; there is more than one definition.

The question about the Frobenius eigenvalues seems prohibitively vague to me: I do not know what it means to "explain" them.

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