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

Abstract

Let I be a prime number and fc an imaginary abelian field. Sinnott [12] has shown that the relative class number of fc is expressed by the so-called index of the Stickelberger ideal of fc, with a "supplementary factor" c~ in N/2 = {n/2\n e N}, and that if fc varies through the layers of the basic Z;-extension over an imaginary abelian field, then c~ becomes eventually constant. On the other hand, c~ can take any value in N/2 as fc ranges over the imaginary abelian fields (cf. [10]). In this paper, we shall study relations between the supplementary factor c~ and Iwasawa's A~-invariant for the basic Zj-extension over fc, our discussion being based upon some formulas of Kida [8,9], those of Sinnott [12], and fundamental results concerning a finite abelian /-group acted on by a cyclic group. As a consequence, we shall see that the A_-invariant goes to infinity whenever fc ranges over a sequence of imaginary abelian fields such that the /-part of c~ goes to infinity.Let Q, R, and C denote the field of rational numbers, the field of real numbers, and that of complex numbers, respectively. A finite abelian extension over Q contained in C will be called, simply, an abelian field. Let k be an imaginary abelian field, namely, an abelian field not contained in R. This field k will be fixed throughout the following sections. We let h~ denote the ratio of the class number of k to that of k+ = k fl R, which is known to be an integer. Let / be a fixed prime number and let Z; denote the ring of /-adic integers. We write fcoo for the basic /(-extension over A: in C.In this paper, we shall discuss relations between Iwasawa's A "-invariant associated with koo and the supplementary factor in an algebraic class number formula for h~ (due to Iwasawa and Sinnott), estimating the /-part of the latter by an elementary function of the former: ord^-R : e-U) < iiAZM_lil(8(A-(fc)-i)/(2(-3) _ 4) if , > 2 <^-(5X-(k)+4)(4x~^+1-1) if/= 2, so that A-(fc)»log(ord,(e-7? : e~U))