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Strong approximation over function fields. (English) Zbl 1428.14035
Let \(C\) be a smooth irreducible projective curve over \(\mathbb{C}\) with function field \(F=\mathbb{C}(C)\). For each place \(v\) of \(F\) , let \(F_v\) be the completion of \(F\) at \(v\). For a nonempty finite set \(S\) of places of \(F\) we denote with \(o_{F,S}\) the ring of \(S\)-integers. Let \(\mathbb{A}_{F,S}:=\prod^\prime_{v \in F \setminus S} F_v\) be the ring of adèles over all places outside \(S\). Here the product considered is restricted, i.e., all but a finite number of factors are in \(o_{F,v}\).
The ring \(\mathbb{A}_{F,S}\) can be equipped with two natural topologies: the first one is the product topology, while the second one is the adelic topology. We consider \(U\) to be a geometrically integral algebraic variety over \(F\). Let \(U(F)\) be the set of \(F\)-rational points, and \(U(\mathbb{A}_{F,S})\) be the restricted product \(\prod^{\prime}_{v \not\in S} U(F_v)\). The set of adelic points \(U(\mathbb{A}_{F,S})\) admits the product topology and the adelic topology as locally inherited from that of adelic affine spaces.
In this context, a strong approximation (respectively, a weak approximation) holds for \(U\) if for any non-empty \(S\) the inclusion \(U(F)\subset U(\mathbb{A}_{F,S})\) is dense in the adelic topology (respectively, in the product topology).
The Hasse’s principle holds for the integral points of \(U\) if for any non-trivial \(S\) and any integral model of \(U\) over \(o_{F,S}\), say \(\mathcal{U}\), \(\prod_{v \not\in S} \mathcal{U}(o_{F,v})\) non-empty implies \(\mathcal{U}(o_{F,S})\) non-empty.
The main aim of this paper paper is to try to build a theory of integral points on open varieties over \(F\). Being expected that weak approximation holds for rationally connected projective varieties, the natural candidates to satisfy strong approximation are log rationally connected varieties, namely varieties on which a general pair of points can be connected by an \(\mathbb{A}_{1}\)-curve.
The main proposed theorem proves that strong approximation holds for smooth, low-degree affine complete intersections and that Hasse’s principle holds for integral points of the interior of any smooth complete intersection pair of low degree.

MSC:
14G05 Rational points
14G25 Global ground fields in algebraic geometry
14M10 Complete intersections
14H05 Algebraic functions and function fields in algebraic geometry
14H10 Families, moduli of curves (algebraic)
14M20 Rational and unirational varieties
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