A local-global principle for torsors under geometric prosolvable fundamental groups. II (Q2668931)

Let \(k\) be a number field, \(X \rightarrow\mathrm{Spec}\, k\) a separated, smooth and geometrically connected curve over \(k\), and \(\eta\) a geometric point of \(X\) with values in its generic point. The point \(\eta\) determines an algebraic closure \(\bar{k}\) of k, and a geometric point \(\bar{\eta}\) of \(X_{\bar{k}} = X \times_{\mathrm{Spec}\,k} \mathrm{Spec}\,\bar{k}\). Let \(\Pi = \pi_1(X,\eta)\) be the arithmetic étale fundamental group of \(X\) with base point \(\eta\), \(\Delta= \pi_1(X_{\bar{k}}, \bar{\eta})\) the étale fundamental group of \(X_{\bar{k}}\) with base point \(\bar{\eta}\), and \(G_k = \mathrm{Gal}(\bar{k}/k)\) the absolute Galois group of \(k\). We suppose that the natural exact sequence \[ 1\rightarrow \Delta \rightarrow \Pi \rightarrow G_k \rightarrow 1 \] splits, and let \(s : G_k \rightarrow \Pi\) be a section of the projection \(\Pi \twoheadrightarrow G_k\). Thus, \(X\) is a \(G_k\)-group via the conjugation action of \(s(G_k)\). Let \(v\) be a prime of \(k\), \(k_v\) the completion of \(k\) at \(v\), and \(G_{k_v} \subset G_k\) a decomposition group associated to \(v\). We view \(\Delta\) as a \(G_{k_v}\)-group via the conjugation action of \(s(G_{k_v})\). Then, we have a natural restriction map (of pointed non-abelian cohomology sets) \(\mathrm{Res}_v : H^1(G_k, \Delta) \rightarrow H_1(G_{k_v}, \Delta)\), and a natural map \[ \prod_{v} \mathrm{Res}_v : H^1(G_k, \Delta) \longrightarrow \prod_v H^1(G_{k_v}, \Delta), \] where the product is over all primes \(v\) of \(k\). This paper deals with the question whether or not the above map is injective. Note that this question is related to the Grothendieck anabelian section conjecture. Let \(\Delta^{\mathrm{sol}}\) be the maximal prosolvable quotient of \(\Delta\). The \(G_k\) (resp. \(G_{k_v}\))-group structure on \(\Delta\) induces a \(G_k\) (resp. \(G_{k_v}\))-group structure on \(\Delta^{\mathrm{sol}}\). Let \(S\) be non-empty subset of the set of primes of \(k\). Then, we have a natural restriction map \[ \prod_{v\in S} \mathrm{Res}_v : H^1(G_k, \Delta^{\mathrm{sol}}) \longrightarrow \prod_{v\in S} H^1(G_{k_v}, \Delta^{\mathrm{sol}}). \] In this paper, it is proved that the above map is injective, provided that the curve \(X\) is affine and the set \(S\) has density 1. This result generalises the author's result in Part I [Manuscr. Math. 145, No. 1--2, 163--174 (2014; Zbl 1297.11066)], where the curve \(X\) is assumed to be proper and \(S\) of density 1. The proof is based on a devissage argument, and a careful analysis of the structure of the geometric prosolvable (resp. geometrically prosolvable arithmetic) (tame) fundamental group of an affine curve.