Explicit families of $K3$ surfaces having real multiplication

Abstract

For families of $K3$ surfaces, we establish a sufficient criterion for real or complex multiplication. Our criterion is arithmetic in nature. It may show, at first, that the generic fibre of the family has a nontrivial endomorphism field. Moreover, the endomorphism field does not shrink under specialisation. As an application, we present two explicit families of $K3$ surfaces having real multiplication by $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{5})$, respectively.