Applications of local-global principles to arithmetic and geometry (Q2715534)

Let \(K\) be a global field and let \(R\) be a Dedekind ring, which is the ring of \(S\)-integers if \(K\) is a number field, \(S\) being a finite set of places of \(K\), or, in the function field case, which is the ring of regular functions outside a finite number of points of the (smooth, complete) curve associated to \(K\). Let \(\widetilde{R}\) be the integral closure of \(R\) in an algebraic closure of \(K\). A theorem of \textit{R. S. Rumely} [J. Reine Angew. Math. 368, 127-133 (1986; Zbl 0581.14014)] says that if \(f:X\to \operatorname {Spec}R\) is an \(R\)-scheme with \(f\) surjective, \(X\) irreducible and \(X_K\) geometrically irreducible, then \(X(\widetilde{R})\neq \emptyset\). This means that (under not too restrictive hypotheses) a finite number of polynomial equations with coefficients in \(R\) have a common solution in \(\widetilde{R}\) if and only if they have a common integral solution locally, at any place of \(K\). The author gives a review of Rumely's local-global principle and its generalizations, one of which is given by the author himself, who has extended it to algebraic stacks [see his forthcoming paper Problèmes de Skolem sur les champs algèbriques, Compos. Math. 125, 1-30 (2001; Zbl 1106.11022)]. Interesting applications are also given, such as constructions of varieties over global fields with prescribed local properties or constructions of Galois extensions of local fields with given groups (among other conditions).NEWLINENEWLINEFor the entire collection see [Zbl 0955.00034].