On the Stickelberger ideal and the relative class number

Abstract

ABSTRACT. Let k be any imaginary abelian field, R the integral group ring of G == Gal(k/Q), and S the Stickelberger ideal of k. Roughly speaking, the relative class number h -of k is expressed as the index of S in a certain ideal A of R described by means of G and the complex conjugation of kj c-h-== [A : S), with a rational number c-in ~N == {n/2;n EN}, which can be described without h-and is of lower than h-if the conductor of k is sufficiently large (cf. [6, 9, 10); see also [5]). We shall prove that 2c-, a natural number, divides 2([k: Q)/2)lk: Q1/2. In particular, if k varies through a sequence of imaginary abelian fields of degrees bounded, then c-takes only a finite number of values. On the other hand, it will be shown that c-can take any value in ~N when k ranges over all imaginary abelian fields. In this connection, we shall also make a simple remark on the divisibility for the relative class number of cyclotomic fields.Let l, Q, IR, and C denote the rational integer ring, the rational number field, the real number field, and the complex number field, respectively. A finite abelian extension over Q contained in C will be called an abelian field. Let k be an imaginary abelian field, namely, an abelian field not contained in IR. We denote by R(k) the group ring of the Galois group G = Gal(k/Q) over l and by s(H), for any subgroup H of G, the sum in R(k) of all elements in H. Putwhere jk denotes the complex conjugation of k. Let h"k denote the relative class number of k (Le., the so-called first factor of the class number of k), Qk the unit index of k, gk the number of distinct prime numbers ramified in k, and S(k) the Stickelberger ideal of k in the sense of Iwasawa-Sinnott, which is an additive subgroup of A(k) with finite index (for the definition of the Stickelberger ideal, see [6,10]). We define c"k as the ratio of the index [A(k) : S(k)] to h"k:The product QkC"k is known to be a natural number and is determined by Sinnott in various cases, for example, in the case gk = lor 2 (cf.[10]). He has also shown in [9] that, if k is a cyclotomic field, then c"k = 2b where b = 0 or 2 gk -1 -1 according as gk = 1 or gk ~ 2 (for the case gk= 1, see [6]).In this paper, we shall give an additional result concerning the range of c"k.

Iwasawa’s 𝜆⁻-invariant and a supplementary factor in an algebraic class number formula

“…We can therefore define a nonnegative integer av = lim ordv(£~R(kn) : £~U(kn)). By this result and by the proof of Lemma 1 in [4] or the remark to Proposition 1 in [10], one can find an upper bound for av, v ^ /, depending only on fc (and /). Our main interest, however, lies in the case v = I.…”

“…On the other hand, (e~R : e~U) is a divisor of ([fc : Q]/2)Ifc:Ql/2, and log[A : S] ~ logh~ as fc ranges over the imaginary abelian fields (cf. [4,10]).…”