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 ${\infty }_K,$ while its base change by $L$ does not so off ${\infty }_L.$ 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.

arXiv admin note: substantial text overlap with arXiv:2010.04919