  A (smooth) K3 surface X defined over a field k of characteristic 0 is called
singular if the N\'eron-Severi lattice NS (X) of X over the algebraic closure
of k is of rank 20. Let X be a singular K3 surface defined over a number field
F. For each embedding \sigma of F into the complex number field, we denote by
T(X^\sigma) the transcendental lattice of the complex K3 surface X^\sigma
obtained from X by \sigma. For each prime ideal P of F at which X has a
supersingular reduction X_P, we define L(X, P) to be the orthogonal complement
of NS(X) in NS(X_P). We investigate the relation between these lattices
T(X^\sigma) and L(X, P). As an application, we give a lower bound of the degree
of a number field over which a singular K3 surface with a given transcendental
lattice can be defined.
