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?



Behaviour of Zeta-function under Finite Morphism
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.


