On the field extension by complex multiplication

Abstract

An abelian variety A with sufficiently many complex multiplications determines over a certain algebraic number field F an abelian extension Kc, namely, the union of all extensions corresponding to ideal sections of A in the sense of the theory of complex multiplication. If we observe KC as a subfield of the maximal abelian extension Ka of F, there arises a problem to investigate which part of Ka is covered by Kc. In the classical case where A is an elliptic curve, it is known that Ka = KC, and there is also a result obtained in [4] for the general case. In the present paper, we shall define for any abelian extension K over F and for any prime number 1 the i-dimension dim, (K/F) of K/F, and show that dim, (K,/F) is, for every 1, equal to a very simple invariant which we shall call the rank of A and denote by rank A. The rank of A depends only on an elementary, group theoretical property of the CM-type to which A belongs, and rank A ? dim A + 1. After the proof of this main result, we shall give an example of nondegenerate abelian variety, i.e., an abelian variety with rank A = dim A + 1. Such an example is given by the Jacobian variety of a hyperelliptic curve of Fermat type. At the end of the paper, we shall add a remark that dim, (KalF) = dim, (Kc/F) holds for some special cases where, among others, a condition about the unit group of F with respect to l is satisfied. This fact suggests that, in many cases, a large part of the maximal abelian extension is obtained by complex multiplication.