I'm trying to get the gist of the proof of the Weil conjectures. Let $X$ be a variety over $\mathbb{F}\_{p^n}$. A priori $Z(X,t)\in \mathbb{Q}[[t]]$. Due to the Grothendieck-Lefschetz fixed point theorem, $Z(X,t)=\prod P\_i(t)^{(-1)^{i+1}}$, where $P\_i(t)$ is the characteristic polynomial of the Frobenius acting on $H^i(X,\mathbb{Q}\_l)$ where $l$ is a fixed prime different from $p$. This implies that $Z(X,t)\in \mathbb{Q}\_l(t)\cap \mathbb{Q}[[t]]$ for every prime $l$ different from $p$.


Does this suffice to determine that it is in $\mathbb{Q}(t)$? If not, then how was it proven that it is in $\mathbb{Q}(t)$?



Is the integrality of the zeta function easy?
It is not necessarily obvious that this suffices, though it is true (fortunately). You can find a proof in Milne's [Lectures on Etale Cohomology](http://jmilne.org/math/CourseNotes/lec.html). (I happened to just be looking at this while thinking about [this question](https://mathoverflow.net/questions/79115/in-what-way-do-the-weil-conjectures-pertain-to-langlands).)


It is Lemma 27.9, phrased as: Let $k\subset K$ be fields and $f(t)\in k[[t]]$. If $f(t)\in K(t)$, then $f(t)\in k(t)$.


