Approximation forte en famille

Abstract

Let $k$ be a number field and $X$ a smooth integral affine variety equipped with a morphism $f:X\to A_k^1$ to the affine line. Assume that all fibres of $f$ are split, for instance that they are geometrically integral. Assume that the generic fibre of $f$ is a homogeneous space of a simply connected, almost simple, semisimple group $G/k(t)$, and that the geometric stabilizers are connected reductive groups. Let $v$ be a place of $k$ such that the fibration $f$ acquires a rational section over the completion $k_v$ at $v$. Assume moreover that at almost all points $x\in A^1(k_v)$ the specialized group $G_x$ is isotropic over $k_v$. If the Brauer group of $X$ is reduced to the Brauer group of $k$, then strong approximation holds for $X$ away from the place $v$.

(July 16th, 2013) The summary is correct, but as Dasheng Wei pointed out to us, there was a mistake in a proof in the previous versions of the paper. Section 3 has been rewritten. Alongside with a technique from earlier papers by the second named author, a new technical tool is Proposition 3.4, which combines strong approximation with a specific proof of Hilbert's irreducibility theorem