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Obstructions to the Hasse principle and weak approximation. (Obstructions au principe de Hasse et à l’approximation faible.) (French) Zbl 1080.14026

Bourbaki seminar. Volume 2003/2004. Exposes 924–937. Paris: Société Mathématique de France (ISBN 2-85629-173-2/pbk). Astérisque 299, 165-193, Exp. No. 931 (2005).
The main object of the paper under review is a smooth, projective, geometrically integral algebraic variety \(X\) defined over a number field \(K\). The author is interested in the following questions: do the Hasse principle and weak approximation hold for certain classes of varieties? If not, is the Brauer–Manin obstruction the only one? As in a series of expository papers by J.-L. Colliot-Thélène, the author’s goal is to present a survey of techniques used for the study of the above problems. The focus is made on the descent method developed by J.-L. Colliot-Thélène and J.-J. Sansuc which is based on the notion of universal torsor. Here the author notices that this notion was rediscovered by T. Delzant in the framework of symplectic geometry and also explains an alternative approach due to D. Cox [J. Algebr. Geom. 4, 17–50 (1995; Zbl 0846.14032)].
Another kind of techniques discussed in the survey is related to the fibration method (note here a variant of Harari’s “formal lemma” whose proof uses a result of T. Graber, J. Harris and J. Starr [J. Am. Math. Soc. 16, 57–67 (2003; Zbl 1092.14063)]). The author also gives a brief review of results due to Skorobogatov and Harari where the Brauer–Manin obstruction is generalized allowing one to explain the counter-examples where it is not the only one. He also explains the relationship with Lang’s conjectures following P. Sarnak and L. Wang [C. R. Acad. Sci., Paris, Sér. I 321, 319–322 (1995; Zbl 0857.14013)] and B. Poonen [in: Rational points on algebraic varieties, Progr. Math. 199, 307–311 (2001; Zbl 1079.14027)]. Finally, the author discusses some generalizations of the Brauer–Manin obstruction: first, from rational points to 0-cycles of degree 1, and second, from number fields to function fields of curves over algebraically closed or real fields.
For the entire collection see [Zbl 1066.00008].

MSC:

14G05 Rational points
14G25 Global ground fields in algebraic geometry
11G35 Varieties over global fields
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