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Embedding problems for open subgroups of the fundamental group. (Problèmes de plongement pour les sous-groupes ouverts du groupe fondamental.) (English. French summary) Zbl 1408.14103
Let \(C\) be a smooth irreducible affine curve over an algebraically closed field \(k\) of characteristic \(p\neq 0\). Abhyankar’s conjecture (proved by M. Raynaud, Invent. Math. 116, No. 1–3, 425–462 (1994; Zbl 0798.14013) and D. Harbater, Invent. Math. 117, No. 1, 1–25 (1994; Zbl 0805.14014)) characterizes the finite quotients of the étale fundamental group \(\pi_1(C)\): a finite group \(G\) is a quotient of \(\pi_1(C)\) if and only if its maximal prime-to-\(p\) quotient is generated by \(2g+r-1\), where \(g\) is the genus of the smooth completion of \(C\) and \(r\) is the number of points in the boundary. But this does not tell how these finite quotients fit together in the inverse system for \(\pi_1(C)\). A natural approach to this question lies in the embedding problem. Recall that a (finite) embedding problem \({\mathcal E}\) for a group \(\Pi\) consists in giving a pair of surjective homomorphisms (\(\alpha:\Pi\to G\), \(\varphi:\Gamma\to G\)), where \(G\) and \(\Gamma\) are finite groups, and looking for a homomorphism \(\gamma:\Pi\to\Gamma\) such that \(\varphi\,\gamma=\alpha\). Given an embedding problem \({\mathcal E}= (\alpha:\pi_1(C)\to G\), \(\varphi:\Gamma\to G)\) and a finite index subgroup \(\Pi\) of \(\pi_1(C)\), let us say that \({\mathcal E}\) restricts to \(\Pi\) if \(\alpha(\Pi)=G\); and if the restricted embedding problem has a solution, that \(\Pi\) is effective for \({\mathcal E}\). A result by D. Harbater and K. Stevenson, Proc. Am. Math. Soc. 139, No. 4, 1141–1154 (2011; Zbl 1231.14019)], states that any finite embedding problem for \(\pi_1(C)\) admits a finite index effective subgroup.
The main goal of this article is to find some necessary and sufficient conditions for a finite index subgroup \(\Pi\) to be effective for a given embedding problem \({\mathcal E}= (\alpha: \pi_1(C)\to G\), \(\varphi:\Gamma\to G\)). The author proceeds by formal patching. Since the statements and their proofs are somewhat intricate, let us be content to say that he combines the techniques of “adding branch points” as by Harbater and Stevenson [loc. cit.] and “increasing the genus” as in his own previous work to produce effective subgroups of \(\pi_1(C)\). Denote \(H=\text{Ker\,}\varphi\) and let \(p(H)\) be the (characteristic) subgroup generated by the \(p\)-subgroups of \(H\).
The main technical part consists in investigating the relationship between solving embedding problems for \(\pi_1(C)\) and what is called the “relative rank” \(\text{rank}_\Gamma(H)\) of the maximal prime-to-\(p\) quotient \(H/p(H)\) in \(\Gamma/p(H)\). Applying this to the case that \(C\) is the affine line and \(\Pi\) has index \(p\) in \(\pi_1(\mathbb{A}^1_k)\), and using degenerations, the author shows that if the curve \(Z\) in the \(\mathbb{Z}/p\mathbb{Z}\) - cover \(Z\to\mathbb{P}^1_k\) corresponding to \(\Pi\subset\pi_1(\mathbb{A}^1_k)\) has genus \(g_Z\) greater than \(\text{rank}_\Gamma(H)\), and under a technical condition which holds for most values of \(g_Z\), \(\Pi\) is effective. Some of the results for the affine line can be extended to general curves although the conclusions are slightly weaker.
14H30 Coverings of curves, fundamental group
14G32 Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory)
12F10 Separable extensions, Galois theory
Full Text: DOI
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