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

TL;DR

This paper investigates how weak approximation with Brauer--Manin obstruction behaves under field extensions, showing that it can fail to be invariant when extending from a number field to a larger one, under certain conjectural assumptions.

Contribution

It demonstrates, assuming Stoll's conjecture, that weak approximation with Brauer--Manin obstruction can fail to be invariant under field extensions, providing explicit examples.

Findings

Existence of a threefold satisfying weak approximation with Brauer--Manin obstruction over K

Failure of this property after base change to an extension L

Explicit unconditional example illustrating the phenomenon

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.