Non-invariance of weak approximation properties under extension of the ground field

TL;DR

This paper investigates how the property of weak approximation with Brauer-Manin obstruction for rational points on algebraic varieties changes when extending the ground field, providing explicit examples and conditions.

Contribution

It demonstrates that weak approximation with Brauer-Manin obstruction is not invariant under field extension, with explicit examples over  and quadratic fields.

Findings

Weak approximation property can fail after field extension.

Explicit example over  showing failure of weak approximation.

Unconditional results for  and certain quadratic fields.

Abstract

For rational points on algebraic varieties defined over a number field $K$, we study the behavior of the property of weak approximation with Brauer-Manin obstruction under extension of the ground field. We construct K-varieties accompanied with a quadratic extension $L/K$ such that the property holds over $K$ (conditional on a conjecture) while fails over $L$. The result is unconditional when $K = \mathbb{Q}$ or $K$ is one of several quadratic number fields. Over $\mathbb{Q}$, we give an explicit example.