Density of rational points on a family of Del Pezzo surfaces of degree 1

Del Pezzo surfaces are geometrically among the easiest surfaces. They have a degree between 1 and 9, and the lower the degree, the more complicated the surface. Those of degree 3 are the famous cubic surfaces containing 27 lines. The arithmetic of Del Pezzo surfaces, however, is much less understood. It is conjectured that on every Del Pezzo surface over a number field k with at least one k-rational point, the set of k-rational points is automatically dense. This has been proved for all Del Pezzo surfaces of degree at least 3, most of degree 2, but only few of degree 1, which is the case we will discuss. These surfaces have a natural elliptic fibration. We will prove for a large family of these surfaces that the set of rational points is Zariski dense if and only if there is at least one fiber (satisfying very mild conditions) that contains infinitely many rational points. This is joint work with Wim Nijgh, based on work of Rosa Winter and Julie Desjardins.