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Mathematics > Number Theory

arXiv:1808.01584 (math)
[Submitted on 5 Aug 2018 (v1), last revised 10 Jul 2019 (this version, v4)]

Title:Brauer-Manin obstruction for Markoff surfaces

Authors:J.-L. Colliot-Thélène, Dasheng Wei, Fei Xu
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Abstract:Ghosh and Sarnak have studied integral points on surfaces defined by an equation x^2+y^2+z^2-xyz= m over the integers. For these affine surfaces, we systematically study the Brauer group and the Brauer-Manin obstruction to the integral Hasse principle. We prove that strong approximation for integral points on any such surface, away from any finite set of places, fails, and that, for m\neq 0, 4, the Brauer group does not control strong approximation.
Comments: This is the final version, to appear in Annali della Scuola Normale di Pisa, classe di Scienze, issue 3 of the XXI/2021 volume, series V, due to be published at the end of September 2021
Subjects: Number Theory (math.NT); Algebraic Geometry (math.AG)
MSC classes: 11G35, 14G25, 11D25, 11D85,
Cite as: arXiv:1808.01584 [math.NT]
  (or arXiv:1808.01584v4 [math.NT] for this version)
  https://doi.org/10.48550/arXiv.1808.01584

Submission history

From: Jean-Louis Colliot-Thélène [view email]
[v1] Sun, 5 Aug 2018 09:06:13 UTC (30 KB)
[v2] Thu, 13 Sep 2018 09:26:36 UTC (39 KB)
[v3] Sat, 8 Dec 2018 10:15:11 UTC (40 KB)
[v4] Wed, 10 Jul 2019 11:31:05 UTC (42 KB)
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