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Harari, David

Fibration method and Manin obstruction. (Méthode des fibrations et obstruction de Manin.) (French) Zbl 0847.14001

Duke Math. J. 75, No. 1, 221-260 (1994).
Let \(k\) be a number field and \(V\) be an algebraic variety defined over \(k\). We have the set \(V(k)\) and the corresponding “local” sets \(V(k_v)\) for any place of \(k\). This gives a possibility to define a Hasse principle and the property of weak approximation for a class of varieties \(V\).
In 1970 Manin introduced an obstruction to the Hasse principle using the Grothendieck-Brauer group and in 1976 Colliot-Thélène and Sansuc constructed an obstruction for the weak approximation. The main problem in this field is to understand that these obstructions are the only ones. In particular, one way to this is to check the validity of this property for fibrations of algebraic varieties if it is true for the base and the fibers. The author proves two theorems of this type (theorem 4.2.1 and 4.3.1).
Reviewer: A.N.Parshin (Moskva)

Cited in 8 Reviews
Cited in 51 Documents

MSC:

14B12 Local deformation theory, Artin approximation, etc.
14F22 Brauer groups of schemes
14G25 Global ground fields in algebraic geometry

Keywords:

fibration; Hasse principle; weak approximation; Grothendieck-Brauer group
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