Non-invariance of Weak Approximation with Brauer–Manin Obstruction

We study weak approximation with Brauer–Manin obstruction with respect to extensions of number fields. For any nontrivial extension [Formula: see text], assuming a conjecture of M. Stoll, we prove that there exists a [Formula: see text]-threefold satisfying weak approximation with Brauer–Manin obstruction off all archimedean places, while its base change to [Formula: see text] fails. Then we illustrate this construction with an explicit unconditional example.