$K3$ surfaces with maximal complex multiplication

Abstract

Let $X$ be a complex projective $K3$ surface having complex multiplication by a CM field $E$, and let $T_X$ be its transcendental lattice. We say that $X$ has maximal complex multiplication if ${\mathrm{End}}_{\mathrm{Hdg}}(T_X)$ is the ring of integers of $E$.
For which CM fields $E$ does such a $K3$ surface exist? What are the possibilities for the transcendental lattices, Picard lattices of these surfaces? The aim of this paper is to study these questions and give some examples.