  Let X be a smooth quartic surface not containing lines, defined over a number
field K. We prove that there are only finitely many bitangents to X which are
defined over K. This result can be interpreted as saying that a certain
surface, having vanishing irregularity, contains only finitely many rational
points. In our proof, we use the geometry of lines of the quartic double solid
associated to X. In a somewhat opposite direction, we show that on any quartic
surface 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.
