Remarks on Brauer-Manin obstruction for Weil restrictions

Abstract

Given a finite extension $K/k$ of number fields and a smooth quasi-projective variety $X$ over $K$. If the abelianized fundamental group of $X$ is trivial, we prove that there is a natural identification between Brauer-Manin sets of $X$ and its Weil restriction $R_{K/k}X$. If $X$ is projective and $Pic(X\times_{K}\overline{k})$ is a torsion-free abelian group, we prove that there is a natural identification between algebraic Brauer-Manin sets of $X$ and $R_{K/k}X$.