ag.algebraic-geometryfinite-fieldsfundamental-group
Clearly the etale fundamental group of $\mathbb{P}^1_{\mathbb{C}} \setminus \{a_1,…,a_r\}$ doesn't depend on the $a_i$'s, because it is the profinite completion of the topological fundamental group. Does the same hold for when I replace $\mathbb{C}$ by a finite field? How about an algebraically closed field of positive characteristic?
(note that I'm talking about the full $\pi_1$ and not the prime-to-$p$ part)
Thanks to Masoud Kamgarpour for bringing this question to my attention; to make up for being 8 years late, I've written way too much. I'll try to (1) clarify the question a bit, (2) list what little explicit knowledge we have about it, and (3) remark on what information we get from the Langlands program.
Let $k$ be a finite field of characteristic $p$ and let $X=\mathbb{P}^1_{k}\setminus\{a_1, \cdots, a_r\},$ where the $a_i\in \mathbb{P}^1_k(k)$.
First, let me try to clarify the question a little bit, as it is not quite well-defined, since the object $\pi_1^{\text{et}}(X_{\bar k})$ is only well-defined up to inner automorphism. To get a well-defined object, one must choose a geometric basepoint $\bar x\in \mathbb{P}^1_k$. Then there is an exact sequence $$1\to \pi_1^{\text{et}}(X_{\bar k}, \bar x)\to \pi_1^{\text{et}}(X, \bar x)\to \text{Gal}(\bar k/k)=\widehat{\mathbb{Z}}\cdot \text{Frob}\to 1.$$
Thus $\text{Gal}(\bar k/k)$ has an outer action on $\pi_1^{\text{et}}(X_{\bar k}, \bar x)$, via conjugation, but to get a well-defined honest action, one needs to choose a splitting of this exact sequence. The action really depends on the choice of splitting.
How does one "choose a splitting"? Natural choices come from choosing the basepoint $\bar x$ to be a rational (or rational tangential) basepoint. That is, let $x\in X$ be a rational point, and let $\bar x$ be the geometric point arising from a choice of algebraic closure of $k$. The map $x\to X$ induces a map $\text{Gal}(\bar k/k)\to \pi_1^{\text{et}}(X, \bar x)$ splitting the exact sequence above, by the functoriality of the etale fundamental group. So we now have an honest action of $\text{Gal}(\bar k/k)$ on $\pi_1^{\text{et}}(X_{\bar k}, \bar x)$, which I claim depends on $x$.
This action is, broadly speaking, quite mysterious. The rest of this answer will aim to explain what is known about it.
First, as the OP explains, it is a result of Grothendieck that (by comparison to the situation over $\mathbb{C}$), the maximal prime-to-$p$ quotient of $\pi_1^{\text{et}}(X_{\bar k}, \bar x)$ is isomorphic to the maximal prime-to-$p$ quotient of $$\langle \gamma_1, \cdots, \gamma_r \mid \prod \gamma_i=1\rangle,$$ that is, the free profinite prime-to-$p$ group on $r-1$ letters. Let $\ell$ be a prime different from $p$; at the cost of losing some information, we will work with the maximal pro-$\ell$ quotient of $\pi_1^{\text{et}}(X_{\bar k}, \bar x)$, which we denote by $\pi_1^\ell$. This is the free pro-$\ell$ group on $r-1$ letters; again the Galois action on this group will depend on the choice of basepoint, though I will suppress it from the notation in the rest of this answer.
Let $$R=\mathbb{Z}_\ell[[\pi_1^\ell]]:=\varprojlim_H \mathbb{Z}_\ell[H],$$ where $H$ runs over the finite quotients of $\pi_1^\ell$ be the group ring of $R$. Instead of describing (what we know about) the action of Frobenius on $\pi_1^\ell$, we will describe its action on $R$; this doesn't lose any information as one can recover $\pi_1^\ell$ as the set of group-like elements in $R$, given the usual Hopf-algebra structure of a group ring. Let $\mathscr{I}\subset R$ be the augmentation ideal. That is, $\mathscr{I}$ is the kernel of the map $R\to \mathbb{Z}_\ell$ sending every group element to $1$. Abstractly, $R$ is isomorphic to a non-commutative power series ring on $r-1$ variables $\mathbb{Z}_\ell\langle\langle X_1, \cdots, X_{r-1}\rangle\rangle$, via the map sending generators $\gamma_i$ of $\pi_1^\ell$ to $(X_i-1)$; under this isomorphism, $\mathscr{I}$ is the two-sided ideal generated by the $X_i$.
First one describes the Galois action on $$\text{gr}_{\mathscr{I}^\bullet} R=\bigoplus \mathscr{I}^n/\mathscr{I}^{n+1}.$$ It is a general fact from the theory of pro-$\ell$ groups that $$\mathscr{I}/\mathscr{I}^2\simeq \pi_1^{\ell, \text{ab}}$$ canonically; hence $$\text{Hom}(\mathscr{I}/\mathscr{I}^2, \mathbb{Z}_\ell)\simeq \text{Hom}(\pi_1^{\ell, \text{ab}}, \mathbb{Z}_\ell)\simeq H^1_{\text{et}}(X_{\bar k}, \mathbb{Z}_\ell)=\mathbb{Z}_\ell(-1)^{\oplus (r-1)}.$$ So Galois acts on $\mathscr{I}/\mathscr{I}^2$ via the cyclotomic character; in other words, Frobenius acts via multiplication by $q$.
But $\mathscr{I}^n/\mathscr{I}^{n+1}\simeq (\mathscr{I}/\mathscr{I}^2)^{\otimes n}$ via the multiplication map, so Frobenius acts on $\mathscr{I}^n/\mathscr{I}^{n+1}$ via multiplication by $q^n$.
So we're done describing the Frobenius action on $\text{gr}_{\mathscr{I}^\bullet} R$; what this tells us is that the interesting data is contained in the extensions between the $\mathscr{I}^n/\mathscr{I}^{n+1}$. More prosaically, we've found that $$\text{Frob}(X_i^n)=q^nX_i^{n}+(\text{terms of degree $>n$}).$$ Giving a good description of these higher-order terms is an open problem, but a lot of work has been done, essentially after replacing $\pi_1^\ell$ with a metabelian quotient thereof. Of course one cannot in practice ``choose generators of $\pi_1^\ell$," so actually computing the coefficients of the terms in the expression above is not a well-stated goal --- but some of their invariants (e.g. the $\ell$-adic valuations of the coefficients) are well-defined enough to make reasonable questions.
Let me briefly summarize some of the literature.
Section 19 of Deligne's classic paper "Le groupe fondamental de la droite projective moins trois points" essentially boils down to computing the action of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on a quotient of $$\mathbb{Z}_\ell[[\pi_1^\ell(\mathbb{P}^1_{\overline{\mathbb{Q}}}\setminus \{0,1\infty\}, \text{tangential basepoint at $0$)}]]\simeq \mathbb{Z}_\ell\langle\langle X, Y\rangle\rangle,$$ where $1+X$ is a loop around $0$ and $1+Y$ is a loop around $1$. Here we quotient by the left ideal generated by $Y$ and all monomials whose $X$-degree is at least 2. If one thinks a bit, one can extract the actions of individual Frobenii on this quotient from his paper. (Deligne doesn't quite phrase things this way, but I find this to be the easiest way to think about what he does.)
Ihara, Nakamura, Wojtkowiak, Anderson and many other authors have performed variants of this computation (in fact, some version of it is originally due to Wojtkowiak, if I remember correctly); Ihara, Nakamura, and Anderson give beautiful symmetric formulae for the Galois action on the quotient of $$G:=\pi_1^\ell(\mathbb{P}^1_{\overline{\mathbb{Q}}}\setminus \{0,1\infty\}, \text{tangential basepoint at $0$})$$ by the normal subgroup generated by $[[G, G],[G,G]]$. Ihara's ICM notes and the references therein are a good summary; one can think of this work as (equivalently) computing the Galois action on the Tate modules of the Jacobians of Fermat curves (which are cofinal among the abelian covers of $\mathbb{P}^1\setminus \{0,1\infty\}$).
The only work I know of that goes substantially beyond metabelian quotients (in approaching these kind of explicit questions), is Anderson-Ihara's paper "Pro-$\ell$ branched coverings of $\mathbb{P}^1$ and higher circular $\ell$-units II." In principle this paper gives an algorithm for computing the power series $$\text{Frob}(X_i^n)=q^nX_i^{n}+(\text{terms of degree $>n$})$$ above, to arbitrary precision (i.e. modulo the ideal $(X_1^N, \cdots, X_{r-1}^N, \ell^N)$, for arbitrary finite $N$). Unfortunately these results are not (at least in my experience) useful for answering theoretical questions about the Galois action. The paper doesn't appear to be available online, but you can find it in the library (or if you can't, I email me and I'll send you a scan).
As far as I know, the state of our knowledge on these precise questions has not progressed dramatically since 1990, when Ihara's ICM notes were written. Of course there has been lots of very interesting incremental progress, e.g. in the beautiful papers of Nakamura and Wojtkowiak; many people (myself included) are still actively thinking about these questions.
That's the summary of explicit work on this issue, at least to my knowledge. After 1990, work on this question has been largely inexplicit, and has focused on exploiting the work of L. Lafforgue on the Langlands program for function fields over $\mathbb{F}_q$ (as well as Deligne's Weil II results). The basic idea is that one can study the outer action of $\text{Gal}(\bar k/k)$ on $\pi_1(X_{\bar k}, \bar x)$ via its action on the set of continuous $\overline{\mathbb{Q}_\ell}$-representations of $\pi_1(X_{\bar k}, \bar x)$; inner automorphisms act trivially on the set of (isomorphism classes of) representations, so this action is actually independent of the basepoint $\bar x$.
Unfortunately it is hard to translate the information that can be extracted from Lafforgue into answers to the sort of explicit questions we asked above. We know:
(In fact, (1) follows from (2) via a sphere-packing argument, which is alluded to in e.g. this paper of Esnault and Kerz, but which as far as I know does not appear in the literature; I can sketch it if you'd like. One can also deduce (1) directly via automorphic methods; Esnault and Kerz sketch this argument.)
This is of course not an exhaustive list; the Langlands program gives much finer information, though it is not necessarily easy to compute with. Much ongoing work aims at bootstrapping these results. For a recent exciting example, see the work of Deligne, Drinfeld, Esnault, Kedlaya, and many others on Deligne's ``companion conjectures" (finally completely proven as of a few days ago) --- see here for a fairly complete reading list on this topic. Of course this is a very active field, and there are other recent papers (my own included) which aim to understand something concrete about $\pi_1$ via the Langlands program.
A final remark: note that properties 1-4 above are purely group-theoretic properties of the outer action of Frobenius on $\pi_1(X_{\bar k})$. In my view, the biggest (if not well-stated) open question about this action is the following:
Can one give a purely group-theoretic proof of 1-4 above? That is, what (explicit) property of the outer action of Frobenius on $\pi_1(X_{\bar k})$ makes 1-4 above true?
For me, at least, this is what it would mean to actually understand the action of Frobenius on the fundamental group.
Best Answer
It is a result of Tamagawa that for two affine curves $C_1, C_2$ over finite fields $k_1,k_2$ any continuous isomorphism $\pi_1(C_1)\rightarrow \pi_1(C_2)$ arises from an isomorphism of schemes $C_1\rightarrow C_2$. Hence, if $\pi_1( \mathbb{P}^1\setminus\{a_1,\ldots, a_r\})$ were independent of the choice of the $a_i$, then the isomorphism class of the schemes $\mathbb{P}^1\setminus\{a_1,\ldots, a_r\}$ would be independent of the choice of $a_1,\ldots,a_r$.
Tamagawa's result is Theorem 0.6 in this paper:
The Grothendieck conjecture for affine curves, A Tamagawa - Compositio Mathematica, 1997 http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=298922
In the case of an algebraically closed field, the answer is also that the fundamental group depends on the choice of the points that are being removed. Again by a theorem by Tamagawa: If $k$ is the algebraic closure of $\mathbb{F}_p$, and $G$ a profinite group not isomorphic to $(\hat{\mathbb{Z}}^{(p')})^2\times \mathbb{Z}_p$, then there are only finitely many $k$-isomorphism classes of smooth curves $C$ with fundamental group $G$ (the restriction on $G$ excludes ordinary elliptic curves).
This can be found in
Finiteness of isomorphism classes of curves in positive characteristic with prescribed fundamental groups, A Tamagawa - Journal of Algebraic Geometry, 2004