Non-invariance of the Brauer-Manin obstruction for surfaces

TL;DR

This paper constructs specific surfaces over number fields demonstrating that the Brauer-Manin obstruction's effectiveness in explaining rational points and the Hasse principle can vary under field extensions, showing non-invariance.

Contribution

It provides explicit constructions of surfaces over number fields that exhibit non-invariance of the Brauer-Manin obstruction properties under field extensions, assuming Stoll's conjecture.

Findings

Surfaces with rational points that lose weak approximation after extension.

Counterexamples to the Hasse principle explained by Brauer-Manin that fail after extension.

Explicit unconditional examples illustrating these phenomena.

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.