Noninvariance of weak approximation with Brauer–Manin obstruction for surfaces
Volume 205 / 2022
Acta Arithmetica 205 (2022), 21-32
MSC: Primary 11G35; Secondary 14G12, 14F22, 14G05.
DOI: 10.4064/aa210827-19-8
Published online: 5 September 2022
Abstract
We study the property of weak approximation with Brauer–Manin obstruction for surfaces with respect to field extensions of number fields. For any nontrivial extension $L/K$ of number fields, assuming a conjecture of M. Stoll, we construct a smooth, projective, and geometrically connected surface over $K$ that satisfies weak approximation with Brauer–Manin obstruction off all archimedean places, while its base change to $L$ does not have this property. We illustrate this construction with an explicit unconditional example.
