Integral points on Markoff type cubic surfaces

Abstract

For integers k, we consider the affine cubic surface $$V_{k}$$ given by $$M(\mathbf{x})=x_{1}^2 + x_{2}^2 +x_{3}^2 -x_{1}x_{2}x_{3}=k$$ . We show that for almost all k the Hasse Principle holds, namely that $$V_{k}({\mathbb {Z}})$$ is non-empty if $$V_{k}({\mathbb {Z}}_p)$$ is non-empty for all primes p, and that there are infinitely many k's for which it fails. The Markoff morphisms act on $$V_{k}({\mathbb {Z}})$$ with finitely many orbits and a numerical study points to some basic conjectures about these "class numbers" and Hasse failures. Some of the analysis may be extended to less special affine cubic surfaces.

Locations

• Inventiones mathematicae - View
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• SHAREOK (University of Oklahoma) - View - PDF
• arXiv (Cornell University) - View - PDF
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• Inventiones mathematicae - View
• arXiv (Cornell University) - View - PDF
• SHAREOK (University of Oklahoma) - View - PDF
• arXiv (Cornell University) - View - PDF
• DataCite API - View
• Inventiones mathematicae - View
• arXiv (Cornell University) - View - PDF
• SHAREOK (University of Oklahoma) - View - PDF
• arXiv (Cornell University) - View - PDF
• DataCite API - View