I am learning the etale fundamental group of a scheme. And I am hugely confused by the etale fundamental group of the additive and multiplicative group scheme $\mathbb{G}_{a}$ and $\mathbb{G}_{m}$ over a finite field $\mathbb{F}_q$.
I know that for an algebraic variety $X$ over a general field $k$, its etale fundamental group suits the following exact sequence $1 \rightarrow \pi_1(X_{k^s}) \rightarrow \pi_1(X) \rightarrow Gal(k^s/k) \rightarrow 1$. So what I want can be captured more or less by $\pi_1(\mathbb{G}_{a,\bar{F_q}})$ and $\pi_1(\mathbb{G}_{m,\bar{F_q}})$.
But the reduced question seems complicated too, because of the characteristic is not $0$. When characteristic $0$ everything is known and can be found, and the result is that $\pi_1(\mathbb{G}_{a,\bar{k}})=1 $ and $\pi_1(\mathbb{G}_{m,\bar{k}})=\mathbb{\widehat{Z}}$, by Riemann-Hurwitz formula or something else. But in characteristic $p$, for $\mathbb{G}_{m}$, at least you don't have the $\mathbb{Z}_p$ part of $\mathbb{\widehat{Z}}$, for the multiplication by $p$ map $\mathbb{G}_m \rightarrow \mathbb{G}_m$ is not unraminfied. And further, the etale fundamental group of $\mathbb{G}_{m}$ seems to be larger than $\prod_{l\neq p}{\mathbb{Z}_l}$, for some kind of higher ramification theory which I don't know.
I also heard the existence of Artin-Schreier extension, but totally don't understand. Can someone give me any tips on that? Or any information about the fundamental group of $\mathbb{G}_{a}$ and $\mathbb{G}_{m}$ over finite field $\mathbb{F}_q$?
Best Answer
Fundamental group in characteristic $p$ consists of a tame part and a wild part. the tame part is computed by the finite etale covers with prime to $p$ degree and basically behave like the situation in characteristic 0. in the case you want the tame part of the fundamental group of $G_a$ is $1$ and for $G_m$ it is $\Pi_{l\not =p}\mathbb{Z}_l$.
there is also a wild part. for this part like the usual algebraic topology it is easier to understand first cohomology. for this write the long exact sequence obtained by evaluating the short exact sequence $$0\to F_p\to O_{G_a}\to O_{G_a}\to 0$$ on $G_a$ where the last map is $Fr-id$, you get $$0\to F_p\to \bar{F_p}[t]\to \bar{F_p}[t]\to H^1(G_a,F_p)\to H^1(G_a,O_{G_a})=0$$ the last equality is true because $G_a$ is affine and hence cohomology of coherent sheafs are zero.
in summary first cohomology of $G_a$ is the cokernel of $Fr-id$. more explicitly if you want an etale cover of degree $p$ you have to choose a polynomial $f(t)\in k(t)$ and then add the root of $x^p-x-f(t)=0$ to $k[t]$. $H^1(G_a,F_p)$ is already a huge group, by a similar trick you can find all the etale covers of degree $p^n$.