Non-invariance of weak approximation with Brauer-Manin obstruction for surfaces

TL;DR

This paper demonstrates that the property of weak approximation with Brauer-Manin obstruction for surfaces can fail to be invariant under field extensions, providing both theoretical and explicit examples.

Contribution

It constructs surfaces that satisfy weak approximation with Brauer-Manin obstruction over a base field but fail over a nontrivial extension, assuming a conjecture of M. Stoll.

Findings

Weak approximation with Brauer-Manin obstruction is not invariant under field extensions.

Explicit unconditional example of such a surface is provided.

Construction relies on a conjecture of M. Stoll.

Abstract

In this paper, we study the property of weak approximation with Brauer-Manin obstruction for surfaces with respect to field extensions of number fields. For any nontrivial extension of number fields L/K, assuming a conjecture of M. Stoll, we construct a smooth, projective, and geometrically connected surface over K such that it satisfies weak approximation with Brauer-Manin obstruction off all archimedean places, while its base change to L fails. Then we illustrate this construction with an explicit unconditional example.