Dino Festi (Mainz)

Density of rational points on diagonal quartic surfaces with two elliptic fibrations

In 2000, Bogomolov and Tschinkel proved that if a K3 surface defined over a number field admits an elliptic fibration, then rational points are potentially dense. About ten years later, Swinnerton-Dyer proves that if a K3 surface X over the rationals has two distinct elliptic fibrations, then there is an explicitly computable closed subset Z such that if X has a rational point outside Z, then rational points are Zariski dense on X. In the same period, Logan, McKinnon, and van Luijk explicitly compute such closed subset for a particular family of K3 surfaces given by diagonal quartic surfaces. In this talk we are going to show that it is possible to extend these results to other families of K3 surfaces.