Strong approximation in families (Q901389)

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 \textit{J.-L. Colliot-Thélène} and \textit{F. Xu} [Acta Arith. 157, No. 2, 169--199 (2013; Zbl 1328.11060)].