\(1\over 2\) Riemann existence theorem with Galois action. (English) Zbl 0840.14012

Frey, Gerhard (ed.) et al., Algebra and number theory. Proceedings of a conference held at the Institute of Experimental Mathematics, University of Essen, Germany, December 2-4, 1992. Berlin: de Gruyter. 193-218 (1994).
Let \(K\) be a field of characteristic \(p\)\(\neq 0\), \(\overline {K}\) be an algebraic closure of \(K\), \(v\) be a rank 1 valuation of \(\overline {K}\) and \(K^h\) be the decomposition field of \(v\) in the separable closure of \(K\) (in \(\overline {K}\)). Let \(S\) be a finite closed subset of \(\mathbb{P}^1_K\) and \(\overline {S}= S\times_K \overline {K}= \{s_1, \dots, s_m\}\) with \(m=2n\) even. The author says that \(S\) is pairwise \(v\)-adjusted if the elements of \(\overline {S}\) can be organized in pairs \(p_k= (x_k, y_k)\), \(1\leq k\leq n\), which are permuted between themselves by \(G^h= \text{Gal} (\overline {K}/ K^h)\) and satisfy \(v(x_k- y_k)> v(x_k- x_{k'})\) for all \(k\neq k'\). Let \(\Pi\) be the profinite group defined (by generators and relations) as follows: \(\Pi= \{g_{x_1}, h_{x_1}, \dots, g_{x_n}, h_{x_n}|g_{x_k} h_{x_k}=1\), \(g^s_{x_k} =1\) for all \(k\}\) where \(s\) is a topological generator of \(\widehat {\mathbb{Z}}\) and \(g^s_{x_k}=1\) is a topological relation; \(\Pi\) is endowed with a right \(G^h\) action by \((\xi\) is the cyclotomic character of \(G^h)\): \(g^\sigma_{x_k}= g^{\xi( \sigma^{-1})}_{\sigma^{-1} x_k}\), \(\sigma\in G^h\).
Main theorem. Let \(\mathbb{U}= \mathbb{P}^1_K- S\), \(\overline {\mathbb{U}}= \mathbb{U}\times_K \overline {K}\), \(\mathbb{U}^{\mathbf h}= \mathbb{U}\times_K K^h\). Suppose that \(S\) is pairwise \(v\)-adjusted for a rank 1 valuation \(v\) of \(\overline {K}\). Then there exists \(\Pi\) as before, an exact sequence of groups with \(G^h\) actions \[ 1\to \Pi\to G^h \propto \Pi\to G^h\to 1 \tag{1} \] which is canonically a quotient of the canonical exact sequence \[ 1\to \pi_1 (\overline {\mathbb{U}})\to \pi_1 (\mathbb{U}^{\mathbf h})\to G^h\to 1. \tag{2} \] Moreover, the generators \(g_{x_k}\) and \(h_{x_k}\) are inertia elements associated to \(x_k\) and \(y_k\) respectively for all \(k\).
Clearly, this result is closed to the classical Riemann existence theorem. It is interesting because \(\text{char} (K)>0\): the canonical sequence (2) is split, but one does not know in general the structure of \(\pi_1 (\overline {\mathbb{U}})\), on the contrary, in the sequence (1), the kernel of \(G^h\propto \Pi\to G^h\) is known. A variant of the main theorem in unequal characteristics gives a “\({1\over 2}\) Riemann existence theorem” (theorem 2).
For the entire collection see [Zbl 0793.00015].


14G20 Local ground fields in algebraic geometry
14E20 Coverings in algebraic geometry
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)