Mathieu Florence: Zéro-Cycles de Degré un sur les Espaces Homogènes

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.

Full text: dvi.gz 27 k, dvi 63 k, ps.gz 655 k, pdf.gz 150 k, pdf 180 k.

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