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Vanishing of Algebraic Brauer-Manin Obstructions


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1 Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv-69978, Israel
     

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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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  • Vanishing of Algebraic Brauer-Manin Obstructions

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Authors

Mikhail Borovoi
Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv-69978, Israel

Abstract


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.