Integral points on Markoff type cubic surfaces

Amit Ghosh; Peter Sarnak

doi:10.1007/s00222-022-01114-z

Integral points on Markoff type cubic surfaces

Amit Ghosh, Peter Sarnak

Mathematics

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5 Scopus citations

Abstract

For integers k, we consider the affine cubic surface V_k given by M(x)=x12+x22+x32-x1x2x3=k. We show that for almost all k the Hasse Principle holds, namely that V_k(Z) is non-empty if V_k(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(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.

Original language	English (US)
Pages (from-to)	689-749
Number of pages	61
Journal	Inventiones Mathematicae
Volume	229
Issue number	2
DOIs	https://doi.org/10.1007/s00222-022-01114-z
State	Published - Aug 2022

All Science Journal Classification (ASJC) codes

General Mathematics

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title = "Integral points on Markoff type cubic surfaces",

abstract = "For integers k, we consider the affine cubic surface Vk given by M(x)=x12+x22+x32-x1x2x3=k. We show that for almost all k the Hasse Principle holds, namely that Vk(Z) is non-empty if Vk(Zp) is non-empty for all primes p, and that there are infinitely many k{\textquoteright}s for which it fails. The Markoff morphisms act on Vk(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.",

author = "Amit Ghosh and Peter Sarnak",

note = "Publisher Copyright: {\textcopyright} 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.",

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T1 - Integral points on Markoff type cubic surfaces

AU - Ghosh, Amit

AU - Sarnak, Peter

PY - 2022/8

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N2 - For integers k, we consider the affine cubic surface Vk given by M(x)=x12+x22+x32-x1x2x3=k. We show that for almost all k the Hasse Principle holds, namely that Vk(Z) is non-empty if Vk(Zp) is non-empty for all primes p, and that there are infinitely many k’s for which it fails. The Markoff morphisms act on Vk(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.

AB - For integers k, we consider the affine cubic surface Vk given by M(x)=x12+x22+x32-x1x2x3=k. We show that for almost all k the Hasse Principle holds, namely that Vk(Z) is non-empty if Vk(Zp) is non-empty for all primes p, and that there are infinitely many k’s for which it fails. The Markoff morphisms act on Vk(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.

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JF - Inventiones Mathematicae

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Integral points on Markoff type cubic surfaces

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