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Strong approximation in families. (Approximation forte en famille.) (French. English summary) Zbl 1334.14014
Let $$X$$ be an algebraic variety over a number field $$k$$ and let $$S$$ be a finite set of places of $$k$$. We have $$X(k)\subset X({\mathbb A}_k)$$. Assume $$X({\mathbb A}_k)\neq \emptyset$$. By definition, $$X$$ satisfies strong approximation off $$S$$ if $$X(k)$$ is dense in $$\text{pr}^S(X({\mathbb A}_k))$$, where $$\text{pr}^S$$ denotes the omission of the $$S$$-components. We also have $$X(k)\subset X({\mathbb A}_k)^{\text{Br}(X)}$$ (elements orthogonal to $$\text{Br}(X)$$ for the Brauer-Manin pairing) and a refinement of the previous definition: $$X$$ satisfies strong approximation with Brauer-Manin obstruction off $$S$$ if $$X(k)$$ is dense in $$\text{pr}^S(X({\mathbb A}_k)^{\text{Br}(X)})$$.
The main result of the present article is a theorem on strong approximation for one-parameter families of homogeneous spaces of linear algebraic groups. Let $$X$$ be a geometrically integral smooth variety over $$k$$ supplied with a $$k$$-morphism $$f$$ from $$X$$ to the affine line with split fibers. Let $$G$$ be a simply connected, almost $$k(t)$$-simple, semisimple group over $$k(t)$$. Assume that the generic fiber of $$f$$ is a homogeneous space of $$G$$ with connected reductive stabilizers. Then strong approximation with Brauer-Manin obstruction is proved for $$X$$ and $$S=\{v\}$$, if $$v$$ is a place of $$k$$ where $$f$$ has a $$k_v$$-rational section and the specialization of $$G$$ at almost all $$t$$ in $$k_v$$ is isotropic over $$k_v$$; moreover, it is assumed that for any $$\alpha\in\text{Br}(X)$$ the evaluation map $$X(k_v)\rightarrow \text{Br}(k_v)$$ determined by $$\alpha$$ is constant. In particular, if $$\text{Br}(X)/\text{Br}(k)=0$$, then strong approximation off $$v$$ holds for $$X$$.
An important part of the proof is a theorem on the specialization of the Brauer group of the generic fiber, extending work by D. Harari. Application of the main theorem to a certain family of quadrics gives a generalization of the results of J.-L. Colliot-Thélène and F. Xu [Acta Arith. 157, No. 2, 169–199 (2013; Zbl 1328.11060)].

##### MSC:
 14G05 Rational points 14G25 Global ground fields in algebraic geometry 11G99 Arithmetic algebraic geometry (Diophantine geometry)
##### Keywords:
strong approximation; Brauer-Manin obstruction
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