Noninvariance 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 $\mathit{L}|\mathit{K}$ such that the property holds over K (conditionally on a conjecture) whereas fails over L. The result is unconditional when K equals $\mathbb{Q}$ or certain quadratic number fields. We give an explicit example when $\mathit{K}=\mathbb{Q}$.