Enriques surfaces with a dense set of rational points

Abstract

The aim of this note is to deduce from the results of Bender and SwinnertonDyer [BS], as refined by Colliot-Thélène [CT], that rational points are Zariski dense on certain Enriques surfaces defined over a number fieldkconditionally on the Schinzel Hypothesis (H) and the finiteness of Tate-Shafarevich groups of elliptic curves overk. It was shown by Bogomolov and Tschinkel [BT] that for any Enriques surface Y defined over a number fieldkthere exists a finite extensionK/ksuch thatK-points are Zariski dense onY(“potential density” of rational points). We intend to show that the results of [BS] and [CT] can be used to construct explicit families of Enriques surfaces over any number fieldkwith the property that alreadyk-points are Zariski dense. Although the general construction is conditional on the above mentioned conjectures, once an equation is written it is often possible to give a direct (and unconditional) proof of the Zariski density of rational points. We check this for the explicit example (4) below, using an idea from [BT]. Let us add that it is not known whether or not there exists an Enriques orK3-surfaceXover a number fieldksuch thatX(k)is not empty but not Zariski dense inX.