Arithmetic of Kummer surfaces and additive combo

Skorobogatov has conjectured that, for K3 surfaces, the Brauer-Manin obstruction is the only obstruction to the local-global principle. Previous works of Colliot-Thélène, Harpaz, Skorobogatov, Swinnerton-Dyer, and others have established this conjecture in some cases if one assumes the finiteness of Sha (and maybe also Schinzel’s hypothesis), but unconditional evidence remains scant. In this talk, I will discuss an ongoing project to prove unconditionally the existence of rational points on certain K3 surfaces (namely, certain geometrically Kummer surfaces) with no Brauer-Manin obstruction. The key idea is to exploit advances in additive combinatorics, à la recent work of Koymans-Pagano. This is in progress work, joint with Katy Woo.