Vanishing of algebraic Brauer-Manin obstructions

Details Vanishing of algebraic Brauer-Manin obstructions Vanishing of algebraic Brauer-Manin obstructions

2010-12-06

arxiv.org/abs/1012.1189v4

Mathematics

Number Theory

12 pages. Final version, to appear in the Journal of the Ramanujan Mathematical Society

Scientific paper

Let X be a homogeneous space of a quasi-trivial k-group G, with geometric stabilizer H, over a number field k. We prove that under certain conditions on the character group of H, certain algebraic Brauer-Manin obstructions to the Hasse principle and weak approximation vanish, because the abelian groups where they take values vanish. When H is connected or abelian, these algebraic Brauer-Manin obstructions to the Hasse principle and weak approximation are the only ones, so we prove the Hasse principle and weak approximation for X under certain conditions. As an application, we obtain new sufficient conditions for the Hasse principle and weak approximation for linear algebraic groups.

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