Formal patching and adding branch points.

*(English)*Zbl 0790.14027The 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 \(G\).

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 \(G\).

Reviewer: N.Yui (Kingston / Ontario)

##### 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 |