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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Author: Kuniaki Horie
Journal: Trans. Amer. Math. Soc. 308 (1988), 313-328
MSC: Primary 11R23; Secondary 11R20, 11R29
MathSciNet review: 946445
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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\vert 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.

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Keywords: Basic $ {\mathbb{Z}_l}$-extension, Iwasawa's $ \lambda $-invariant, abelian field, Stickelberger ideal, class number
Article copyright: © Copyright 1988 American Mathematical Society

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