Iwasawa’s $\lambda ^ -$-invariant and a supplementary factor in an algebraic class number formula

Abstract:

Let $l$ be a prime number and $k$ an imaginary abelian field. Sinnott [12] has shown that the relative class number of $k$ is expressed by the so-called index of the Stickelberger ideal of $k$, with a "supplementary factor" ${c^ - }$ in $\mathbb {N}/2 = \{ n/2|n \in \mathbb {N}\}$, and that if $k$ varies through the layers of the basic ${\mathbb {Z}_l}$-extension over an imaginary abelian field, then ${c^ - }$ becomes eventually constant. On the other hand, ${c^ - }$ can take any value in $\mathbb {N}/2$ as $k$ ranges over the imaginary abelian fields (cf. [10]). In this paper, we shall study relations between the supplementary factor ${c^ - }$ and Iwasawa’s ${\lambda ^ - }$-invariant for the basic ${\mathbb {Z}_l}$-extension over $k$, our discussion being based upon some formulas of Kida [8, 9], those of Sinnott [12], and fundamental results concerning a finite abelian $l$-group acted on by a cyclic group. As a consequence, we shall see that the ${\lambda ^ - }$-invariant goes to infinity whenever $k$ ranges over a sequence of imaginary abelian fields such that the $l$-part of ${c^ - }$ goes to infinity.