Quartic surfaces, their bitangents and rational points

Let X be a smooth quartic surface not containing lines, defined over a numberfield K. We prove that there are only finitely many bitangents to X which aredefined over K. This result can be interpreted as saying that a certainsurface, having vanishing irregularity, contains only finitely many rationalpoints. In our proof, we use the geometry of lines of the quartic double solidassociated to X. In a somewhat opposite direction, we show that on any quarticsurface X over a number field K, the set of algebraic points in X(\overeline K)which are quadratic over a suitable finite extension K' of K is Zariski-dense.Comment: This is the final version of the pape