On the connected components of moduli of real polarized K3-surfaces

We complete the investigations in [11] on the classification of connected components of moduli of real polarized K3-surfaces. In particular, we show that this classification is closely related to some classical problems in number theory: the classification of binary indefinite lattices and the representation of integers as sums of two squares. As an application, we use recent results in [13] to completely classify real polarized K3-surfaces that are deformations of real hyperelliptically polarized K3-surfaces. This is important because real hyperelliptically polarized K3-surfaces can be constructed explicitly.