Manin obstruction to strong approximation for homogeneous spaces

TL;DR

This paper proves a strong approximation theorem for adelic points on homogeneous spaces over number fields, extending previous results by incorporating the Manin obstruction and Brauer groups.

Contribution

It generalizes strong approximation results to broader classes of homogeneous spaces using the Brauer-Manin obstruction, including transcendental elements.

Findings

Strong approximation holds for adelic points orthogonal to certain Brauer group subgroups.

Extends previous results by Harari and Colliot-Thélène–Xu to more general homogeneous spaces.

Utilizes the Brauer-Manin pairing and existing theorems in the proof.

Abstract

For a homogeneous space X (not necessarily principal) of a connected algebraic group G (not necessarily linear) over a number field k, we prove a theorem of strong approximation for the adelic points of X in the Brauer-Manin set. Namely, for an adelic point x of X orthogonal to a certain subgroup (which may contain transcendental elements) of the Brauer group Br(X) of X with respect to the Manin pairing, we prove a strong approximation property for x away from a finite set S of places of k. Our result extends a result of Harari for torsors of semiabelian varieties and a result of Colliot-Th\'el\`ene and Xu for homogeneous spaces of simply connected semisimple groups, and our proof uses those results.