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Strong approximation for the total space of certain quadric fibrations. (English) Zbl 1328.11060
Let $$X$$ be a variety over a number field $$F$$ such that the set $$X(F)$$ of rational points is nonempty. Let $$S$$ be a finite set of places in $$F$$. The strong approximation for $$X$$ off the set $$S$$ means that the diagonal image of $$X(F)$$ is dense in the space of $$S$$-adéles $$X({\mathbb A}_{F}^{S}).$$ The $$S$$-adéles are the adéles with places in $$S$$ omitted. If the property of strong approximation holds for $$X$$ then it implies a local-global principle for the existence of integral points on models over the ring of $$S$$-integers of $$F.$$ There is an obstruction called the Brauer-Manin obstruction to strong approximation off $$S.$$
In this paper, the authors investigate strong approximation for varieties $$X/F$$ given by an equation $q(x_{1},\dots ,x_{n}) = p(t),$ where $$q$$ is a quadratic form of rank $$n$$ in $$n\geq 3$$ variables and $$p(t)$$ is a nonzero polynomial.

##### MSC:
 11G05 Elliptic curves over global fields 14G05 Rational points 14F22 Brauer groups of schemes
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