@article{a2d1077594914bc59859c6b167944da4,

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 = "Funding Information: We thank V. Blomer, E. Bombieri, J. Bourgain, T. Browning, C. McMullen, P. Whang and U. Zannier for insightful discussions. AG thanks the Institute for Advanced Study and Princeton University for making possible visits during part of the years 2015-2017 when much of this work took place. He also acknowledges support from the IAS, a Simons Foundation grant No. 634846 and his home Department. He dedicates this article to his family Priscilla, Armand and Saskia. PS was supported by NSF grant DMS 1302952. The softwares Eureqa and Mathematica were used on a PC running Linux to generate some of the data. Additional computations were done at the OSU-HPCC at Oklahoma State University, which is supported in part through the NSF grant OCI-1126330. We also thank the referees for suggestions that improved the paper. Publisher Copyright: {\textcopyright} 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.",

year = "2022",

month = aug,

doi = "10.1007/s00222-022-01114-z",

language = "English (US)",

volume = "229",

pages = "689--749",

journal = "Inventiones Mathematicae",

issn = "0020-9910",

publisher = "Springer New York",

number = "2",

}