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Noninvariance of weak approximation with Brauer–Manin obstruction for surfaces

Volume 205 / 2022

Han Wu Acta Arithmetica 205 (2022), 21-32 MSC: Primary 11G35; Secondary 14G12, 14F22, 14G05. DOI: 10.4064/aa210827-19-8 Published online: 5 September 2022


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.


  • Han WuHubei Key Laboratory of Applied Mathematics
    Faculty of Mathematics and Statistics
    Hubei University
    No. 368, Friendship Avenue, Wuchang District
    Wuhan, Hubei, 430062, P.R. China

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