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On some Hasse principles for algebraic groups over global fields. III. (English) Zbl 1420.11072

Summary: We establish some new local-global principles related with some splitting problems for connected linear algebraic groups over infinite algebraic extensions of global fields and give some applications to the isotropy problems. The main tools are certain new Hasse principles established for quadratic, (skew-)hermitian forms, and homogeneous projective spaces of reductive groups over such fields.
For Parts I and II see [Proc. Japan Acad., Ser. A 90, No. 5, 73–78 (2014; Zbl 1300.11032); 90, No. 8, 107–112 (2014; Zbl 1310.11043).

MSC:

11E72 Galois cohomology of linear algebraic groups
14F20 Étale and other Grothendieck topologies and (co)homologies
14L15 Group schemes
14G20 Local ground fields in algebraic geometry
20G10 Cohomology theory for linear algebraic groups
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