Fields of definition of K3 surfaces with complex multiplication.

Let X / C be a K3 surface with complex multiplication by the ring of integers of a CM field E. We show that X can always be defined over an Abelian extension K / E explicitly determined by the discriminant form of the lattice NS (X). We then construct a model of X over K via Galois-descent and we study some of its basic properties, in particular we determine its Galois representation explicitly. Finally, we apply our results to give upper and lower bounds for a minimal field of definition for X in terms of the class number of E and the discriminant of NS (X). [ABSTRACT FROM AUTHOR]