Integral points on Markoff type cubic surfaces

Amit Ghosh, Peter Sarnak

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

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’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
Volume229
Issue number2
DOIs
StatePublished - Aug 2022

All Science Journal Classification (ASJC) codes

  • General Mathematics

Fingerprint

Dive into the research topics of 'Integral points on Markoff type cubic surfaces'. Together they form a unique fingerprint.

Cite this