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The defect of strong approximation in connected linear groups. (Le défaut d’approximation forte dans les groupes linéaires connexes.) (French. English summary) Zbl 1216.11047
Summary: 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.

##### MSC:
 11E72 Galois cohomology of linear algebraic groups 14L15 Group schemes 14G05 Rational points
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