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

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.