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Zbl 0704.14014
Moret-Bailly, Laurent
Groupes de Picard et problèmes de Skolem. I. (Picard groups and Skolem problems. I). (French)
[J] Ann. Sci. Éc. Norm. Supér. (4) 22, No. 2, 161-179 (1989). ISSN 0012-9593

A problem of Skolem asks about the existence of a solution of a system of diophantine equations over a ring R of algebraic integers whose coordinates belong to a finite extension of R. In the scheme-theoretical language this is a question about the existence of a multi-section of a morphism of finite type $f:X\to Spec(R)$, where R is a Dedekind ring. \par The main result of this paper asserts that the answer is positive if the following conditions are satisfied: \par (i) R is an excellent ring whose residue fields are algebraic over a finite field; \par (ii) for any finite extension $K'$ of the field of fractions K of R the normalization $R'$ of R in $K'$ has torsion Picard group; \par (iii) there exists $K'$ as above such that one of the irreducible components of the base change $X\otimes\sb RR'$ is mapped surjectively to $Spec(R').$ \par This result is a generalization of a theorem of {\it R. S. Rumely} [J. Reine Angew. Math. 368, 127-133 (1986; Zbl 0581.14014)], who assumed that R is a ring of integers in an algebraic number field and f is surjective. The proof is geometric and does not use the theory of capacity of Rumely. \par [See also the following review.]
[I.V.Dolgachev]

MSC 2000:: *14G25 Global ground fields
13F05 Dedekind and Pruefer rings and their generalizations
11R04 Algebraic numbers
11D41 Higher degree diophantine equations

Keywords: Skolem problems; finite extension ring of algebraic integers; existence of a solution of a system of diophantine equations

Citations: Zbl 0704.14015; Zbl 0581.14014

Cited in: Zbl 1101.14306 Zbl 1106.11022 Zbl 1014.11040 Zbl 0924.14023 Zbl 0863.14016 Zbl 0704.14015

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