Integral points on Markoff type cubic surfaces

Abstract

For integers k, we consider the affine cubic surface V k given byWe 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.

Order 5 Brauer–Manin obstructions to the integral Hasse principle on log K3 surfaces

“…The Hasse principle and the effectivity of the Brauer-Manin obstruction for the equation x 3 + y 3 + z 3 − xyz = k were studied by Ghosh and Sarnak [13], Colliot-Thélène, Wei and Xu [8], and Loughran and Mitankin [21]. Another classical affine cubic equation was studied in this manner by Bright and Loughran [4].…”