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# Structure of homogeneous spaces and applications to local-global principles

## Cyril Demarche

(joint work with Giancarlo Lucchini-Arteche) We study the structure of homogeneous spaces of linear algebraic groups over perfect fields and prove a result reducing certain nice properties or conjectures about homogeneous spaces to the crucial case of homogeneous spaces of the special linear group with finite stabilizers. In particular, one can apply this reduction to the Brauer-Manin obstruction to the Hasse principle and weak approximation for rational points or zero-cycles of degree one over number fields (as in a recent work by Harpaz and Wittenberg).