MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.)

share|cite|improve this question

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.