# Behaviour of Zeta-function under Finite Morphism

Let X ---> Y be a finite surjective morphism of smooth, projective, connected varieties over a finite field F_q. Can one describe the zeta function Z(X, t) in terms of the zeta-function Z(Y,t) of Y? Can one say anything at all about how they are related? What if we assume the morphism is finite etale?

-
Brandon: I would accept one of the answers below. Hypothetically both, except that that's not possible. –  Greg Kuperberg Dec 7 '09 at 7:56

I don't think you can say much of anything of consequence unless you have better control over things. As an example, consider the kth power map A^1 -> A^1. This is finite, and surjective and even etale if you throw out 0, and has degree k, but the zeta functions are the same.

Another way of rerephrasing Ilya's comment is that it's not just varieties at have zeta functions; all mixed sheaves have them. The zeta function of X is the zeta function of the pushforward of the constant sheaf on X to Y. Now it may be that you can say something useful about this sheaf (for example, if you have a cover, it is a local system, and you can think about L-functions of pi_1 representations) but in general, that sheaf could be pretty horrible.

-
Still, in the case you're able to establish some results about L-functions, like in the number theory, this is a pretty useful identity! –  Ilya Nikokoshev Oct 27 '09 at 20:11
That's true. It just tends to be the sort of thing where conservation of trouble kicks in. Working with zeta functions of varieties over finite fields directly is easy enough that it's not clear that thinking about L-functions will tell you much new. –  Ben Webster Oct 27 '09 at 21:25

You should be looking not at just zeta functions, but at the L-functions.

Then yes, for a finite etale Galois morphism the identity should be

        Z(Y) = Z(X) * L(X, \pi_1) * L(X, \pi_2) * ...


(where the product is over summands of the regular representation of Galois group of the morphism, Z being the L-function of the trivial representation.) This is in no way restricted to finite fields — in fact the idea as well as the notation comes from theorem about Dirichlet L-functions.

The proof is that by a definition of what is L-function it can be written either for a trivial mixed sheaf on Y (LHS) or for its pushforward on X (RHS).

-
To make my answer simpler, I was looking for simple introduction into what L-function is from this point of view but I don't think I know any. Anyone has some reference to recommend? –  Ilya Nikokoshev Oct 27 '09 at 20:28

Let $E\to C$ be an elliptic surface (say defined over $\mathbb{Q}$. Then Mike Rosen and I proved a formula for the rank of the group of sections $C\to E$ defined over $\mathbb{Q}$ in terms of the average number of points modulo $p$ on the fibers. (The result is conditional on Tate's conjecture for the surface $E$.) Anyway, the proof is via a comparison of the zeta function of $E$ as a surface compared to the zeta functions of the fibers and the zeta function of the base curve $C$. So not a direct relation that you seem to be asking for, but it has the flavor of your question. The paper is

M. Rosen, J.H. Silverman, On the rank of an elliptic surface, Invent. Math. 133 (1998), 43-67.

There have since been some extensions to 1-dimensional families of abelian varieties and/or to higher dimensional bases, which you can find by forward referencing our article on MathSciNet.

-

The paper by Aubry and Perret Divisibility of zeta functions of curves in a covering, Archiv der Math. 82(3) (2004), 205-213 contributes to this question. A free version is available at link text.

-