Invariance Of Brauer-Manin Obstruction

In the first half of this paper,we construct three kinds of Chatelet surfaces,which have some given arithmetic properties with respect to field extensions of number fields.We then use these constructions to study the properties of weak approximation with Brauer-Manin obstruction and the Hasse principle with BrauerManin obstruction for 3-folds with respect to field extensions of number fields.Let L/K be a nontrivial extension of number fields.We assume a conjecture of M.Stoll.We construct a 3-fold defined over K,which has a K-rational point,and satisfies weak approximation with Brauer-Manin obstruction off ∞ K,while its base change by L does not so off ∞L.For the Hasse principle with Brauer-Manin obstruction,assuming additionally either the odd degree of the field extension[L:K]or the existence of a real place for L,we construct a 3-fold defined over K,which is a counterexample to the Hasse principle explained by the BrauerManin obstruction,while the failure of the Hasse principle of its base change by L cannot be so.Then we illustrate these constructions and some exceptions with some explicit unconditional examples.In the second half of this paper,we study the properties of weak approximation with Brauer-Manin obstruction and the Hasse principle with Brauer-Manin obstruction for surfaces with respect to field extensions of number fields.We also assume the conjecture of M.Stoll.For any nontrivial extension of number fields L/K,we construct two kinds of smooth,projective,and geometrically connected surfaces defined over K.For the surface of the first kind,it has a K-rational point,and satisfies weak approximation with Brauer-Manin obstruction off ooK,while its base change by L does not so off ∞ L.For the surface of the second kind,it is a counterexample to the Hasse principle explained by the Brauer-Manin obstruction,while the failure of the Hasse principle of its base change by L cannot be so.Then we illustrate these constructions with explicit unconditional examples.