Brauer-Manin sets of X over K and its Weil restriction R_{K/k}X are naturally identified when the abelianized fundamental group of X is trivial or when X is projective with torsion-free Picard group.
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Remarks on Brauer-Manin obstruction for Weil restrictions
Brauer-Manin sets of X over K and its Weil restriction R_{K/k}X are naturally identified when the abelianized fundamental group of X is trivial or when X is projective with torsion-free Picard group.