Calculate the zeta function of a scheme by from its étale covers?

Question

The title says it all. Given a (proper smooth) scheme $X$ over Spec$\mathbb{F}_q$, is it possible to calculate the zeta function of $X$ by from its étale covers?

Like for $\mathbb{P}^n$ you can cover it by $(n+1)$ copies of $\mathbb{A}^n$ which has as $k$-fold intersection of the form $\mathbb{G}_m^k \times \mathbb{A}^{n-k}$, and counting points are like doing the Inclusion-exclusion principle. Now I'm wondering if it can be done in the étale topology.

More concretely, given a scheme and a finite étale cover of it, how are their zeta functions related?

(BTW, I know there is an easier way to calculate the zeta function of $\mathbb{P}^n$ by using an affine stratification, the above example is just to illustrate my question.)

Answer 1

If $Y \to X$ is an etale galois cover with group $G$ (all over a finite field $\mathbb{F}_q$) then, for each $g \in G$ you can consider the twist $Y^{(g)} \to X$ which is also an etale cover of $X$. (Twists are parametrized by elements of $G$ rather that an $H^1$ because Galois groups of extensions of finite fields are cyclic). Now $X( \mathbb{F}_q)$ is the disjoint union of the images of the $Y^{(g)}( \mathbb{F}_q)$, so you get get the zeta function of $X$ from that of the $Y^{(g)}$'s. With some extra care over the ramified points, the same can be done for any galois (not necessarily etale) cover.