La descente sur les variétés rationnelles. II. (The descent on rational varieties. II)
[For part I see Journées de géometrie algébrique, Angers/France 1979, 223-237 (1980; Zbl 0451.14018).] The article under review gives a general framework for the study of rational points on varieties X over number fields k. For such a variety the Hasse-principle holds if it has a k-rational point, provided that it has rational points over all local fields \(k_ v\) arising from completions). Yu. I. Manin has found an obstruction to the Hasse- principle, involving Brauer-groups. The authors observe that one can give a geometric interpretation by forming torsors \(E\to X\) (under tori), such that the Manin-obstruction vanishes for E. Any rational point on X lifts to one such torsor E, and it is hoped that E satisfies the Hasse- principle. This is true for many important examples. However these assertions have been shown elsewhere. The bulk of this paper consists of general and very detailed cohomological constructions and computations.
scientific article; zbMATH DE number 4077431
scientific article