Integral points on Markoff type cubic surfaces

Amit Ghosh, Peter Sarnak

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)689-749
Number of pages61
JournalInventiones Mathematicae
Issue number2
StatePublished - Aug 2022

All Science Journal Classification (ASJC) codes

  • General Mathematics


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