Approximation forte en famille

TL;DR

This paper establishes conditions under which strong approximation holds for certain algebraic varieties over number fields, particularly those with fibers that are homogeneous spaces of semisimple groups, using Brauer group conditions and local-global principles.

Contribution

It provides new criteria for strong approximation on affine varieties with homogeneous space fibers, linking Brauer groups and local isotropy conditions.

Findings

Strong approximation holds away from a specified place under given conditions.

The Brauer group of the variety is trivial over the base field.

Conditions on fibers ensure the validity of strong approximation.

Abstract

Let $k$ be a number field and $X$ a smooth integral affine variety equipped with a morphism $f : X \to A^1_k$ 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 $v$.