Summary
Letk be a perfect field with cdk(i)≤1. It has recently been proved by the author that homogeneous spaces under connected linear groups overk satisfy a Hasse principle with respect to the real closures ofk. Using this result we classify the semisimple algebraic groups overk and, in particular, characterize the anisotropic ones. Similarly we classify the various types of hermitian forms over skew fields overk and exhibit to what extent weak or strong local-global principles hold. In the case wherek is the function field of a smooth projective curveX over ℝ, we also cover the local-global questions vis-à-vis the completions ofk at the points ofX.
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Scheiderer, C. Classification of hermitian forms and semisimple groups over fields of virtual cohomological dimension one. Manuscripta Math 89, 373–394 (1996). https://doi.org/10.1007/BF02567524
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DOI: https://doi.org/10.1007/BF02567524