Remarks on Brauer-Manin obstruction for Weil restrictions

TL;DR

This paper investigates the relationship between Brauer-Manin obstructions and Weil restrictions of varieties over number fields, establishing natural identifications under certain conditions.

Contribution

It proves natural correspondences between Brauer-Manin sets of a variety and its Weil restriction, extending understanding of obstructions in arithmetic geometry.

Findings

Brauer-Manin sets of X and R_{K/k}X are naturally identified when the abelianized fundamental group of X is trivial.

Algebraic Brauer-Manin sets of X and R_{K/k}X are identified when X is projective and Picard group is torsion-free.

The results connect obstructions for varieties over different fields via 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 $R_{K/k}X$.