Curves in $\overline{\Bbb{F}}_p$

Question

Let $k$ an algebraically closed field, for example $k = \Bbb{C}$ or $k = \overline{\Bbb{F}}_p$, $a \in k$, $k(x)$ the field of rational functions, fix an algebraic closure $\overline{k(x)}$,

let $R_a$ be a subring of $\overline{k(x)}$ which is henselian at $a$, that is $(x-a)$ is a maximal ideal and for a polynomial $P \in R_a[T]$, if $P(c) = 0 \bmod (x-a), P'(c) \ne 0 \bmod (x-a)$ then $c$ lifts to a unique root $\in R_a$ of $P$.

Concretely one can start from a realization of $\overline{k(x)}$ as Puiseux series at $a$ and make a subring $R_a$ from the elements whose Puiseux series is a power series.

Pick another such subring $R_b\subset \overline{k(x)}$, then $R=R_a \cap R_b$ can be thought as an abstract curve $\gamma : a \to b \subset k$.

For $f \in R $, from the reductions $\bmod (x-a)^n$, we have the expansions $$f = \sum_{n \ge 0} c_n (x-a)^n \in \varprojlim R/(x-a)^n\\ f = \sum_{n \ge 0} d_n (x-b)^n \in \varprojlim R/(x-b)^n $$ and the map $\sum_{n \ge 0} c_n (x-a)^n \mapsto \sum_{n \ge 0} d_n (x-b)^n $ is the analytic continuation along $\gamma$.

Questions : Does this construction of abstract curves works, is there a better one, how to define closed loops $a\to a$, can we obtain a group from them, can we get homotopy classes of curves when restricting to a ring (which one ?) smaller than $\overline{k(x)}$, is it possible to give examples of abstract curves in $\overline{\Bbb{F}}_p$ ?

In $\Bbb{C}$ we don't have only the curves in the complex topology, but also things like this : let $\gamma_n$ be the circle of radius $r \to 1^+$ traversed $n$ times, then analytic continuation along $\gamma_n$ is well-defined for any algebraic function analytic at $z=1$, and $\gamma_n$ should be an abstract curve even for $n \in \hat{\Bbb{Z}}$ (the profinite integers).

We should have an isomorphism between such closed loops $1 \to 1$ in $\Bbb{C}$ and $Gal(\overline{\Bbb{C}(x)}/\Bbb{C}(x))$.

Answer 1

Your approach seems related to Grothendieck's notion of an abstract path ("chemin") via isomorphisms of fiber functors, as in SGA 1. There are other approaches to the notion of "path" in abstract situations, like Coleman's theory of $p$-adic integration and Voevodsky's $\mathbb A^1$ homotopy theory---but those are quite different.

In your situation, given $a \in k$, the ring $R_a$ is not unique as a subring of $\bar K$, where $K = k(x)$. The possible choices correspond to the valuations on $\bar K$ that extend the $(x-a)$-adic valuation on $K$. Another way to say this is to let $A$ be the integral closure of $k[x]$ in $\bar K$; then the choices of $R_a$ correspond to the closed points on $\text{Spec}\,A$ over $(x-a)$ (i.e. the maximal ideals of $A$ lying over $(x-a)$). To define a loop at $x=a$, you would make two choices of $R_a$; or equivalently, two valuations on $\bar K$ over the $(x-a)$-adic valuation on $K$; or again equivalently, two maximal ideals of $A$ lying over $(x-a)$. If the two picks are the same, then you would have the trivial loop.

Your goal seems to be to define and understand "fundamental groups" in this framework, and to relate that to absolute Galois groups. But for that, you would need to say what the branch locus is. This relates to what you wrote at the end of your question, which was not quite right.

For example, if $k$ is the complex numbers, then the loops (usual or profinite) along the unit circle around the origin correspond to the elements of the (topological or étale) fundamental group of the affine line with the origin removed; and this is $\mathbb Z$ (respectively $\hat{\mathbb Z}$). This is also the Galois group over $\mathbb C(x)$ of the maximal subfield of $\overline{\mathbb C(x)}$ that is unramified away from $0$ and infinity (obtained by adjoining the $m$th root of $x$ for all $m$).

If you instead delete $n$ points from the affine line (i.e. allowing $n$ specified points to be branch points) then you will get more covers, and a bigger fundamental group, which will be the (discrete or profinite) free group on $n$ generators. You would then get the Galois group of the pro-universal cover of the $n$-punctured affine line, taken at some base point $x=a$. Choosing a second base point, and points on the pro-universal cover over those two points, is roughly what Grothendieck calls a "chemin" (though he would start with a closed point and a generic point downstairs).

Above, if you allow unrestricted branching, then in the limit you will get the free (discrete or profinite) group on an infinite generating set corresponding to the points of the affine line (i.e. the complex numbers). This corresponds to the fact that the absolute Galois group of $\mathbb C(x)$ is free profinite on that many generators. This can be thought of as coming from taking a loop around every point of the affine line, since every point is allowed to ramify when passing to the algebraic closure of $\mathbb C(x)$.

The situation over an algebraically closed field $k$ of characteristic $p$ is more involved. There, it is also true that the absolute Galois group of $k(x)$ is free profinite, on a set of generators of cardinality equal to that of $k$. But the reason is different. That is because if you delete $n$ points from the affine $k$-line, then the corresponding étale fundamental group is not a free profinite group. But amazingly, the inverse limit of those profinite groups is free (as can be shown using embedding problems; this is a theorem proven by David Harbater and by Florian Pop). It would be interesting to know what the generators of that free group correspond to, since it is not the set of points of the affine $k$-line.

Update:

You gave me a lot of clues, for the $\mathbb C$ part I can sketch some arguments for most aspects (the main tool I have is that with enough analytic continuations of $f$ then the coefficients of $\prod (T - f_\gamma)$ are locally meromorphic with trivial monodromy thus they are meromorphic on the Riemann sphere thus they are rational functions).

From the fundamental group $\pi_1(X)$ of $X = \mathbb{C} - \{a_1, ... , a_n\}$ I get that if $F/\mathbb{C}(z)$ is Galois generated by algebraic functions locally analytic away from the $a_j$ then $\text{Gal}(F/\mathbb{C})$ is a quotient of $\pi_1(X)$, thus the profinite completion of $\pi_1(X)$ is the Galois group of the maximal field unramified away from the $a_j$ and infinity, and it is the étale fundamental group of X. In characteristic p instead of $\bar k(x)$ for the base field I would try to take $\bar k(x, x^{1/p^\infty})$ (because its algebraic closure is separable).

About the paragraph about loop = pairs of valuations: an isomorphism $\overline{k(x)} \to k((x - a))[(x - a)^{1/\infty}]$ (the Puiseux series) which is the natural map on $k(x)$ is the same as a $\mathbb Q$-valued function on $\overline{k(x)}$ above the one of $k[x]_{(x - a)}$ plus a choice of (coherent) sequence $\pi_n = (x - a)^{1/n}$ in $\overline{k(x)}$ (we can't fix $(\pi_n)_n$ one for all because otherwise our pair of valuations will only induce automorphisms that are trivial on $k(x)[(x - a)^{1/\infty}]$).

So once we have fixed an isomorphism to the Puiseux series at $a$, a closed loop $a \to a$ = an element of the Galois group = an other isomorphism to the Puiseux series at $a$ = one valuation above $x - a$ plus an element of $\hat{\mathbb{Z}}^\times$.

What you wrote about loops seems reasonable. But there is the question of what space you are taking your loops in, since you are taking loops at a point that is allowed to branch (which is ordinarily not allowed in topology). One interpretation might be that from a topological point of view you are taking tangential base points, in the sense of Grothendieck's dessins d'enfants. For example, if one takes covers of the Riemann sphere branched at $0,1,\infty$, then one can take a tangential base point at $0+$, a point "just to the right of $0$ but infinitely close", or alternatively at $0$ but starting out by moving in the direction of the point $1$. That way, starting at $0$ and going counterclockwise around $0$ before returning to $0$ (from the right) makes sense as a non-trivial loop. This ties in with the Grothendieck–Teichmüller group $\widehat{GT}$.

Let me make a couple of additional comments.

In characteristic $p$, it is traditional to take the separable closure rather than the algebraic closure, and therefore not to take $p$-power roots of $x$. On the other hand, taking all those roots is related to what happens with perfectoid spaces.

Although you are focusing on loops due to the connection with absolute Galois groups and fundamental groups, you could also do something similar with paths, using Grothendieck's idea of fundamental groupoids, which are in a way more natural (even if less classical). These are discussed in SGA 1 and elsewhere. This perspective was used, for example, in the last several pages of Harbater-Hartmann-Krashen-Parimala-Suresh's recent paper "Local-global Galois theory of arithmetic function fields" https://arxiv.org/abs/1710.03635 (where there is a citation to a book [Br06] by a topologist on that viewpoint).