Transcendental Lattices and Supersingular Reduction Lattices of a Singular K3 Surface

A (smooth) K3 surface X defined over a field k of characteristic 0 is called singular if the Néron-Severi lattice NS(X) of X ⊗ k is of rank 20. Let X be a singular K3 surface defined over a number field F . For each embedding σ : F →֒ C, we denote by T (Xσ) the transcendental lattice of the complex K3 surface Xσ obtained from X by σ. For each prime ideal p of F at which X has a supersingular reduction Xp, we define L(X, p) to be the orthogonal complement of NS(X) in NS(Xp). We investigate the relation between these lattices T (Xσ) 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.