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Formal patching and adding branch points. (English) Zbl 0790.14027
The paper is concerned with the fundamental group $$\pi_ 1$$ of an affine curve in characteristic $$p$$. The prime-to-$$p$$ part of $$\pi_ 1$$ is well understood, but the full group $$\pi_ 1$$ remains still mysterious. S. Abhyankar [Am. J. Math. 79, 825-856 (1957; Zbl 0087.036)] conjectured that a finite group $$G$$ is a quotient of $$\pi_ 1$$ if and only if every prime-to-$$p$$ quotient of $$G$$ is a quotient of $$\pi_ 1$$ of an analogous curve in characteristic 0. This conjecture was proved in the affirmative in several cases: Grothendieck for $$G$$ itself a prime-to-$$p$$ group; Serre over the affine line for $$G$$ solvable, and Abhyankar himself obtained a related result using the technique of “enlarging” Galois groups.
In this paper, the author proves that $$G$$ is a quotient of $$\pi_ 1$$ if a certain subgroup of $$G$$ is. This done by introducing a new technique of “patching for formal deformation of curves” and “adding new branch points on a given cover”. This technique strengthens the method developed by the author in the series of earlier papers, where “mock covers” were used.
Theorem A. Let $$G$$ be a finite group, with subgroups $$H$$, $$H_ 1,\dots,H_ r$$; $$H_ 1',\dots,H_ s'$$ $$(s\leq r)$$, where $$H_ i\subset H_ i'$$ for $$i\leq s$$, each $$H_ i'$$ is a $$p$$-group, and all $$H_ i\subset H$$. Let $$X$$ be an irreducible regular $$k$$-curve, and let $$Y\to X$$ be a regular $$H$$-Galois cover that is unramified outside the set $$B=\{\gamma_ 1,\dots,\gamma_ r\}\subset X$$. Suppose that $$H_ i$$ arises as an inertia group over $$\gamma_ i$$ for each $$i$$. Then the following assertions hold:
(a) There is a regular $$G$$-Galois cover $$Z\to X$$ unramified outside $$B$$, such that $$H_ i'$$ (resp. $$H_ i)$$ arises as an inertia group of a point of $$Z$$ over $$\gamma_ i$$ for $$i\leq s$$ (resp. for $$i>s)$$.
(b) One may take $$Z$$ to be irreducible provided that $$Y$$ is irreducible and $$G$$ is generated by $$H$$, $$H_ 1',\dots,H_ s'$$.
As an application of theorem A, realizations of covering groups are discussed.
Assume from here on that $$k$$ is an algebraically closed field of characteristic $$p$$.
Theorem B. Let $$X$$ be an irreducible regular projective $$k$$-curve, and let $$G$$ be a finite group. Then the following assertions hold:
(a) $$G$$ is a Galois group of an irreducible regular Galois branched cover of $$X$$.
(b) Suppose that $$G$$ is generated by $$p$$-subgroups $$P_ 1,\dots,P_ m$$ together with elements $$h_ 1,\dots,h_ r$$ having orders prime to $$p$$. Then the cover in (a) may be chosen to have at most $$2r+m$$ branch points.
(c) Let $$g$$ be the genus of $$X$$. In (b), if $$h_ 1,\dots,h_ r$$ generate a prime-to-$$p$$ subgroup $$H$$ of $$G$$, then the cover in (a) may be chosen to have at most $$s$$ branch points, where $$s=m$$ if $$r\leq g$$, $$s=m+1$$ if $$g<r\leq 2g$$, and $$s=r+m+1-2g$$ if $$r>2g$$. Moreover, the positions of these branch points may be chosen arbitrary.
Corollary. Let $$G$$ be a finite group generated by two elements $$g$$, $$h$$, where $$g$$ has a $$p$$-power order, while $$h$$ has prime-to-$$p$$ order. Then $$G$$ is a quotient of $$\pi_ 1(C)$$ for every smooth connected affine $$k$$- curve $$C$$, with possible exception of curves isomorphic to $$A^ 1$$ or $$A^ 1-\{0\}$$.
Specializing $$G$$ and $$p$$, some finite groups are realized as Galois groups over all affine $$k$$-curves except possibly $$A^ 1$$ or $$A^ 1- \{0\}$$: this list includes the Monster $$M$$ with $$p=2,3,29$$ or 71; $$PSL_ 2(2^ n)$$ with $$p=2$$, or $$p=2^ n+1$$; $$M_{11}$$, $$M_{23}$$, $$J_ 1$$, etc. with appropriate choices of $$p$$.
The detailed study of covers of particular curves, e.g., curves of genus $$\leq 1$$ with one point deleted is carried out.
Proposition. Let $$E$$ be an elliptic curve over $$k$$, and let $$e$$ be a point $$E$$, and let $$G$$ be a finite group. Then $$G$$ is a quotient of $$\pi_ 1(E-\{e\})$$ provided either
(a) $$G$$ is generated by a $$p$$-subgroup $$P\subset G$$ together with an element $$h\in G$$; or
(b) $$G$$ is a quasi-$$p$$-group of order $$p(p+1)s$$ where $$s$$ is an integer none of whose non-trivial factors is congruent to 0 or 1 modulo $$p$$.
Again specializing $$G$$ and $$p$$, some finite groups are realized as Galois groups over all one-punctured ordinary elliptic curves in characteristic $$p$$: this list includes $$M_{12}$$ with $$p=2$$ or 3; $$PSL_ 2(F_ r)$$ with $$p=r>5$$, etc. – Finally, for genus zero case, the conjecture of Abhyankar is affirmatively proved.
Theorem C. Let $$G$$ be a finite group, and let $$H$$ be a subgroup of $$G$$ such that $$p|(G:K)$$ for all proper subgroups $$K\subset G$$ containing $$H$$. If $$H$$ is a quotient of $$\pi_ 1(A^ 1)$$, then so is $$G$$.

##### MSC:
 14H30 Coverings of curves, fundamental group 14G15 Finite ground fields in algebraic geometry 12F12 Inverse Galois theory 14F35 Homotopy theory and fundamental groups in algebraic geometry 20D15 Finite nilpotent groups, $$p$$-groups 20F29 Representations of groups as automorphism groups of algebraic systems
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