Rational points on del Pezzo surfaces of degree 1 and root numbers of elliptic curves

A theorem of Segre and Manin states that if a del Pezzo surface X over Q of degree 2 or higher has a rational point that misses the exceptional curves of X, then the set of rational points on X is Zariski dense. Del Pezzo surfaces of degree 1 (dP1s) carry a canonical rational point, but Zariski density of rational points in this case has remained a mystery. The purpose of this talk is to shed some light on this problem.

Blowing up the canonical rational point of a dP1 we obtain an elliptic fibration over P¹(Q). Using sieving techniques, we will show that, for a large class of fibrations corresponding to dP1s, there is always an infinite family of fibers all of whose members have negative root number. Assuming finiteness of certain Tate-Shafarevich groups, this shows Zariski Density of rational points on the corresponding dP1 surfaces.

If time permits, we will also discuss some surprising fibrations that correspond to dP1 surfaces, all of whose fibers have positive root number.