Submission: 2004, Feb 26
We explain how to build homogeneous spaces of a connected linear algebraic group, having zero-cycles of degree one but no rational point, even on a smooth compactification. Roughly, those spaces are obtained by applying a nonabelian cohomological machinery to a given extension of finite groups which is non-split, but split over every $p$-Sylow of the base. We give two precise examples: the first and easiest one, over the field C(x))(y)); the second one, which requires a little more work, over a local or global field, and with finite abelian stabilizers. Both are rational varieties.
2000 Mathematics Subject Classification: 14L30
Keywords and Phrases: algebraic group, homogeneous space, zero-cycle, galois cohomology, nonabelian cohomology.
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