Question

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 $\mathbb{Q}(t)$?

Answer 1

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. (I happened to just be looking at this while thinking about this question.)

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 $f(t)\in k(t)$.

Answer 2

First embedd $\mathbf{Q}_{\ell}\subseteq\mathbf{C}$ and then there is quite an easy argument that I discovered which answers your question, see the following link 1.

In fact this argument shows that for any normal subfield $K\subseteq\mathbf{C}$, if $$ \frac{f}{g}\in\mathbf{C}(x_1,x_2,\ldots,x_n)\cap K[[x_1,x_2,\ldots,x_n]] $$ with $\gcd(f,g)=1$ and the constant term of $f$ and $g$ are equal to $1$ then $f,g\in K[x_1,\ldots,x_n]$.