## $$1\over 2$$ 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].

### MSC:

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