Chen, Sheng Integral points on twisted Markoff surfaces. (English) Zbl 1468.11149 J. Number Theory 220, 212-234 (2021). Summary: We study the integral Hasse principle for affine varieties of the shape \[ a x^2 + y^2 + z^2 - x y z = m, \] using the Brauer-Manin obstruction, and we produce examples whose Brauer groups include 4-torsion elements. We describe these elements explicitly, and in some cases, we show that there is no Brauer-Manin obstruction to the integral Hasse principle for them. MSC: 11G35 Varieties over global fields 11D25 Cubic and quartic Diophantine equations 14F22 Brauer groups of schemes 14G12 Hasse principle, weak and strong approximation, Brauer-Manin obstruction Keywords:Brauer group; Brauer-Manin obstruction; group cohomology; integral point; Hasse principle PDFBibTeX XMLCite \textit{S. Chen}, J. Number Theory 220, 212--234 (2021; Zbl 1468.11149) Full Text: DOI arXiv References: [1] Bright, M. J., Obstructions to the Hasse principle in families (2016) [2] Colliot-Thélène, J.-L.; Xu, F., Brauer-Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms, Compos. Math., 145, 2, 309-363 (2009), with an appendix by D. Wei and F. Xu · Zbl 1190.11036 [3] Colliot-Théléne, J.-L.; Kanevsky, D.; Sansuc, J.-J., Arithmétique des surfaces cubiques diagonales, (Diophantine Approximation and Transcendence Theory. Diophantine Approximation and Transcendence Theory, Lecture Notes in Math., vol. 1290 (1987), Springer: Springer Berlin) · Zbl 0639.14018 [4] Colliot-Thélène, J.-L.; Wei, D.; Xu, F., Brauer-Manin obstruction for Markoff surfaces [5] Fulton, W., Intersection Theory (1998), Springer: Springer Berlin · Zbl 0885.14002 [6] Gille, P.; Szamuely, T., Central Simple Algebras and Galois Cohomology, Cambridge Studies in Advanced Mathematics, vol. 101 (2006), Cambridge University Press · Zbl 1137.12001 [7] Harshorne, R., Algebraic Geometry, GMT, vol. 52 (1977), Springer [8] Loughran, D.; Mitankin, V., Integral Hasse principle and strong approximation for Markoff surfaces This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.