Integral points on Markoff type cubic surfaces

• 成果类型: Article
• 署名作者: Ghosh, Amit; Sarnak, Peter
• 署名单位: Oklahoma State University System; Oklahoma State University - Stillwater; Institute for Advanced Study - USA; Princeton University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-022-01114-z
• 发表日期: 2022
• 页码: 689-749
• 关键词: hasse principle; density; form

摘要

For integers k, we consider the affine cubic surface V-k given by M(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(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.