Rational and integral points on Markoff-type K3 surfaces

Time: 9:30 -- 11: 00, April 16, 2025

Venue: Room 612, A6, Institute of Mathematics-VAST

Abstract: Following a recent work of Fuchs, Litman, Silverman, and Tran on Markoff-type K3 (MK3) surfaces, we study the local-global principle for integral points on these surfaces. In particular, we construct a family of MK3 surfaces with infinitely many rational points but no integral points due to the Brauer–Manin obstruction, and give some counting results of a similar nature to those in recent works of Ghosh and Sarnak; Loughran and Mitankin; and Colliot-Thélène, Wei, and Xu. We also give results on the computation of geometric Picard groups and algebraic Brauer groups of MK3 surfaces in this family.