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Prime-to-\(p\) étale fundamental groups of punctured projective lines over strictly Henselian fields. (English) Zbl 1439.14100

Let \(R\) be a strictly Henselian DVR of characteristic 0 and residue characteristic \(p\). Set \(K=\text{Frac}(R)\) and let \(\bar{K}\) be an algebraic closure of \(K\). Let \(\alpha_1,\ldots,\alpha_d\in\mathbb{P}_K^1(K)\) and let \(P\in\mathbb{P}_K^1(\bar{K})\smallsetminus \{\alpha_1,\ldots,\alpha_d\}\). This paper is concerned with the maximal prime-to-\(p\) quotient of the étale fundamental group of \(\mathbb{P}^1_K\smallsetminus\{\alpha_1,\ldots,\alpha_d\}\), denoted \(\pi_1^{\text{ét}}(\mathbb{P}^1_K\smallsetminus\{\alpha_1,\ldots,\alpha_d\},P)^{(p')}\). The absolute Galois group \(G_K\) of \(K\) acts on this group, giving a homomorphism \[ \rho_{\text{alg}}^{(p')}:G_K\longrightarrow\text{Aut} (\pi_1^{\text{ét}}(\mathbb{P}^1_K\smallsetminus \{\alpha_1,\ldots,\alpha_d\},P)^{(p')}). \] Let \(\bar{\rho}_{\text{alg}}^{(p')}\) be the homomorphism from \(G_K\) to the outer automorphism group of \(\pi_1^{\text{ét}}(\mathbb{P}^1_K\smallsetminus \{\alpha_1,\ldots,\alpha_d\},P)^{(p')}\) induced by \(\rho_{\text{alg}}^{(p')}\).
In this paper, the authors give an explicit description of \(\pi_1^{\text{ét}}(\mathbb{P}^1_K\smallsetminus\{\alpha_1,\ldots,\alpha_d\},P)^{(p')}\) in terms of generators and relations, and determine the homomorphism \(\bar{\rho}_{\text{alg}}^{(p')}\). When \(\alpha_i\in R\cup\{\infty\}\) for \(1\le i\le d\) they also determine \(\rho_{\text{alg}}^{(p')}\). Their descriptions depend only on the intersection multiplicities \(E_{ij}\) of the divisors on \(\mathbb{P}_R^1\) given by \(\alpha_i\) and \(\alpha_j\) for \(1\le i<j\le d\).
The proofs are based on considering a parallel problem in topology. Let \(\epsilon>0\), let \(B_{\epsilon}\) be the open disc of radius \(\epsilon\) centered at 0 in the complex plane, and set \(B_{\epsilon}^*=B_{\epsilon}\smallsetminus\{0\}\). Let \(\mathcal{F}\) be a family of copies of \(\mathbb{P}_{\mathbb{C}}^1\) with \(d\) points removed, parameterized by \(B_{\epsilon}^*\). The points which are removed from each copy of \(\mathbb{P}_{\mathbb{C}}^1\) are given by convergent power series \(a_1(x),\ldots,a_d(x)\) on \(B_{\epsilon}\). Thus the space associated to \(z\in B_{\epsilon}^*\) is \(\mathcal{F}_z=\mathbb{P}_{\mathbb{C}}^1\smallsetminus \{a_1(z),\ldots,a_d(z)\}\). Choose a base point \(z_0\in B_{\epsilon}^*\); then the point at infinity on \(\mathcal{F}_{z_0}\) serves as a base point for \(\mathcal{F}\), which we denote by \(\infty_{z_0}\). Let \(\hat{\pi}_1(B_{\epsilon}^*,z_0)\) be the profinite completion of \(\pi_1(B_{\epsilon}^*,z_0)\), and let \(\hat{\pi}_1(\mathcal{F},\infty_{z_0})^{(p')}\) be the maximal prime-to-\(p\) quotient of the profinite completion of \(\pi_1(\mathcal{F},\infty_{z_0})\). There is an action of \(\hat{\pi}_1(B_{\epsilon}^*,z_0)\) on \(\hat{\pi}_1(\mathcal{F},\infty_{z_0})^{(p')}\), and hence a homomorphism \[ \rho_{\text{top}}^{(p')}:\hat{\pi}_1(B_{\epsilon}^*,z_0)\longrightarrow\text{Aut}(\hat{\pi}_1(\mathcal{F},\infty_{z_0})^{(p')}). \] The authors show that \(\rho_{\text{alg}}^{(p')}\) factors through \(G_K^{(p')}\), and that \(\rho_{\text{top}}^{(p')}\) factors through \(\hat{\pi}_1(B_{\epsilon}^*,z_0)^{(p')}\). Suppose the series \(a_1(x),\cdots,a_d(x)\) satisfy \(v_x(a_i(x)-a_j(x))=E_{ij}\) for \(1\le i<j\le d\). The main result of this paper says that there are isomorphisms \(\hat{\pi}_1(B_{\epsilon}^*,z_0)^{(p')} \cong G_K^{(p')}\) and \[ \hat{\pi}_1(\mathcal{F},\infty_{z_0})^{(p')}\cong\pi_1^{\text{ét}}(\mathbb{P}^1_K\smallsetminus \{\alpha_1,\ldots,\alpha_d\},P)^{(p')} \] inducing an isomorphism of the outer actions \(\bar{\rho}_{\text{top}}^{(p')}\) and \(\bar{\rho}_{\text{alg}}^{(p')}\). The authors then use topological methods to determine \(\bar{\rho}_{\text{top}}^{(p')}\), which leads to a description of \(\bar{\rho}_{\text{alg}}^{(p')}\).

MSC:

14H30 Coverings of curves, fundamental group
14G20 Local ground fields in algebraic geometry
11S20 Galois theory
14G32 Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory)
14F35 Homotopy theory and fundamental groups in algebraic geometry
57M05 Fundamental group, presentations, free differential calculus
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
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