On degenerate CM-types (Q1342383)

Let \(K\) be a CM-field of degree \(2d\) and let \(\Gamma_ K\) denote the set of embeddings of \(K\) into \(\mathbb{C}\). A subset \(S\) of \(\Gamma_ K\) is called a CM-type if \(S \bigcup S\rho = \Gamma_ K\) and \(S \bigcap S\rho = \emptyset\), where \(\rho\) denotes the complex conjugation. The rank of \(S\) was defined by \textit{T. Kubota} [Trans. Am. Math. Soc. 118, 113-122 (1965; Zbl 0146.279)] in order to study the abelian extensions obtained by complex multiplication. Let \(A\) be an abelian variety with complex multiplication of type \((K,S)\). Then \(\dim A = d\) and \(\text{rank }S \leq d + 1\). If \(\text{rank }S < d + 1\), then \(S\) is called degenerate. The author gives a method of constructing degenerate CM-types systematically. He also investigates the fields of moduli for such CM-types.