Non-invariance of the Brauer-Manin obstruction for surfaces
Abstract
In 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 assume a 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 ∞K, 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. We illustrate these constructions with explicit unconditional examples.