Given a smooth affine curve $U$ over an algebraically closed field $k$, we wish to describe the algebraic fundamental group $\pi_1(U)$, a nd its set of finite quotients $\pi_A(U)$. Here $U=X-S$, where $X$ is a smooth projective curve of genus $g\geq 0$, and $S=\{\xi_0,\cdots\xi_r\}$ $(r\geq 0)$. If $k={\bf C}$, then by topology we know that $\pi_1(U)$ is the free profinite group on $2g+r$ generators, and $\pi_A(U)$ consists of finite groups generated by $2g+r$ elements. But this does not hold in characteristic $p$, e.g. because of Artin-Schreier covers of the affine line.
In 1957, Abhyankar [Ab1] posed:
Abhyanksar's Conjecture ("AC"). In
characteristic $p$, if $X$ has genus $g$ and $U=X-\{\xi_0,\cdots,
\xi_r\}$, then a finite group $G$ is in $\pi_A(U)$ if and only if
every prime-to-$p$ quotient of $G$ has $2g+r$ generators.
Later results of Grothendieck [Gr] imply that the forward implication
is true; that a prime-to-$p$ group $G$ is in $\pi_A(U)$ if a nd only
if it has $2g+r$ generators; and thos not all groups conjectured to be
in $\pi_A(U)$ can arise form branched covers of $X$ that are tamely
ramified over $\{\xi_0,\cdots, \xi_r\}$ and unramified elsewhere. This
suggests the following:
Strong Abhyankar Conjecture ("SAC"). In
characteristic $p$, if $U=X-\{\xi_0,\cdots, \xi_r\}$ with $X$ of genus
$g$, and if each prime-to-$p$ quotient of $G$ lies in $\pi_A(U)$,
then $G$ is the Galois group of a Galois étale cover of $U$ whose
smooth completion is tamely ramified over $X$ except possibly at
$\xi-0$.
As a result of recent work of Raynaud and the speaker, we now have
Theorem. [Ha3] SAC (and hence AC) holds for all
affine curves in characteristic $p$.
This determines $\pi_A(U)$, but the profinite group $\pi_i(U)$ remains unknown, and little is known about what kinds of inertia groups can occur over the points $\xi_i\in S$. Also, AC shows that $\pi_A(U)$ depends only on the values of $g$ and $r$; but this is false for $\pi_1$ in characteristic $p$ [Ha4]. The situation is quite different for projective curves in characteristic $p$, wheree $\PI_A$ is smaller than it is for complex curves of the same genus; where even $\pi_A$ depends on the choice of curve (and not just on the genus); and where there is not even a conjecture about what $\pi_A$ is when $g>1$ (although it is now known [St] that if $G$ has $g$ generators, and $g'\geq g$, then $G\in \pi_A(X)$ for a generic $X$ of genus $g'$). And if instead we consider branched covers without any restrictions on the positions of the branch points, then the fundamental group (i.e. the absolute Galois group of the function field) is a free profinite group [Ha5] [Po].
Although Abhyankar's Conjecture was stated in 1957, evidence began to accumulate only about 1980. The case of $U={\bf A}^1$ (where SAC=AC) was considered first; there AC says that $\pi_A$ consists of the quasi-$p$-groups (i.e. groups generated by their $p$-subgroups). In this situation, Nori (cf. [Ka]) and Abhyankar (cf. [Ab2]) showed that various finite groups, especially certain simple groups, lie in $\pi_A({\bf A}^1)$. Later, Serre [Se] used a cohomological argument to show that AC over ${\bf A}^1$ holds for solvable quasi-$p$-groups. The full AC for ${\bf A}^1$ was shown by Raynaud [Ra] in 1992, using a patching argument in rigid analysis.
An analogous form of patching had been used by Harbater [Ha2] in 1991, to obtain partial results toward SAC for general affine curves. Here formal geometry was used, especially a "formal GAGA" patching result. (The formal patching theorem in [Ha2] strengthened earlier patching results that had allowed the speaker to prove in [Ha1] that every finite group is a Galois group over ${\bf Q}_p(x)$.) In 1993, the results and methods of [Ha2] were combined with Raynaud's result to obtain the full SAC [Ha3].
In both rigid and formal patching, the main idea is to "cut and paste" curves in characteristic $p$ much as one would with Riemann surfaces. To do this one needs more open sets than in the Zariski topology -- and the rigid and formal settings each provide ways of accomplishing this. To construct covers of a curve $U$ with Galois group $G$, one may construct covers of simpler curves with simpler Galois groups, and then paste these together so as to construct the desired cover. Here, it is possible to work inductively. Group theory is used in order to build up $G$ from smaller groups, and algebraic geometry and ring theory are needed to construct the smaller covers and the patching data.
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