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Groupes de Picard et problèmes de Skolem. II. (Picard groups and Skolem problems. II). (French) Zbl 0704.14015
The author proves the following refinement of the main theorem of part I of this paper [ibid. 22, No.2, 161-179 (1989; see the preceding review)] the notation of which we preserve. Assume that R is global, i.e. arises as a localization of a ring of algebraic numbers K or as an open affine set of a smooth algebraic curve over a finite field. Suppose we are given additionally a finite set \(\Sigma\) of valuations of K which do not come from maximal ideals of R, for each \(v\in \Sigma\) we fix a finite Galois extension \(L_ V\) of the completion \(K_ V\) of K and a non-empty open subset \(\Omega_ V\) of \(X(L_ V)\) whose points are smooth and invariant with respect to \(Gal(L_ V/K_ V)\). Then, assuming that \(f:X\to Spec(R)\) is surjective and the union of \(\Sigma\) and the maximal spectrum of R is not equal to the set of all places of K, one can find an irreducible closed subscheme Y of X finite over R such that for any \(v\in \Sigma\), \(Y\otimes_ RL_ V\) consists of \(L_ V\)-rational points contained in \(\Omega_ V.\)
The case \(\Sigma =\emptyset\) corresponds to a theorem of Rumely (see part I). The present geometric proof specializes in this case to the proof from part I.
Reviewer: I.V.Dolgachev

MSC:
14G25 Global ground fields in algebraic geometry
11R04 Algebraic numbers; rings of algebraic integers
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
11D41 Higher degree equations; Fermat’s equation
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References:
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