Le d\'efaut d'approximation forte dans les groupes lin\'eaires connexes

TL;DR

This paper investigates the failure of strong approximation in connected linear algebraic groups over number fields, linking it to the algebraic Brauer group and establishing exact sequences that clarify the obstructions to rational and integral points.

Contribution

It introduces an exact sequence describing the closure of rational points in adelic points, connecting the defect of strong approximation to the algebraic Brauer group, and extends Poitou-Tate sequences to non-abelian Galois cohomology.

Findings

The algebraic Brauer-Manin obstruction is the only obstruction to integral points on torsors under G.

Established a non-abelian Poitou-Tate exact sequence for Galois cohomology.

Described the closure of rational points in adelic points via an exact sequence.

Abstract

Let G be a connected linear algebraic group over a number field k. We establish an exact sequence describing the closure of the group G(k) of rational points of G in the group of adelic points of G. This exact sequence describes the defect of strong approximation on G in terms of the algebraic Brauer group of G. In particular, we deduce from those results that the integral Brauer-Manin obstruction on a torsor under the group G is the only obstruction to the existence of an integral point on this torsor. We also obtain a non-abelian Poitou-Tate exact sequence for the Galois cohomology of the linear group G. The main ingredients in the proof of those results are the local and global duality theorems for complexes of k-tori of length two and the abelianization maps in Galois cohomology introduced by Borovoi.