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

Abstract

In this paper, we study weak approximation with Brauer--Manin obstruction with respect to extensions of number fields. For any nontrivial extension $L/K,$ assuming a conjecture of M. Stoll, we prove that there exists a $K$-threefold satisfying 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.

arXiv commentThis is a part of our paper "Châtelet surfaces and non-invariance of the Brauer--Manin obstruction for 3-folds" arXiv:2010.04919. The part on "the Hasse principle with Brauer-Manin obstruction" is being considered by the editor of manuscripta mathematica. So we reorganize the rest part, and simplify the construction of Châtelet surfaces to make it more readable