Fibration method and Manin obstruction
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).
scientific article; zbMATH DE number 706711
scientific article