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Châtelet surfaces and non-invariance of the Brauer-Manin obstruction for $3$-folds (2010.04919v2)

Published 10 Oct 2020 in math.NT

Abstract: In this paper, we construct three kinds of Ch^atelet surfaces, which have some given arithmetic properties with respect to field extensions of number fields. We then use these constructions to study the properties of weak approximation with Brauer-Manin obstruction and the Hasse principle with Brauer-Manin obstruction for $3$-folds, which are pencils of Ch^atelet surfaces parameterized by a curve, with respect to field extensions of number fields. We give general constructions (conditional on a conjecture of M. Stoll) to negatively answer some questions, and illustrate these constructions and some exceptions with some explicit unconditional examples.

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