Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Ch\^atelet surfaces and non-invariance of the Brauer-Manin obstruction
  for $3$-folds   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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The categories of this paper are: acc-phys
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The categories of this paper are: funct-an
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The categories of this paper are: hep-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: patt-sol
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The categories of this paper are: solv-int
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The categories of this paper are: physics
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-ex
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The categories of this paper are: cs
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The categories of this paper are: q-bio
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The categories of this paper are: astro-ph
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The categories of this paper are: supr-con
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The categories of this paper are: chao-dyn
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The categories of this paper are: gr-qc
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The categories of this paper are: adap-org