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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



On the $ l$-adic Iwasawa $ \lambda$-invariant in a $ p$-extension

Authors: Eduardo Friedman and Jonathan W. Sands
Journal: Math. Comp. 64 (1995), 1659-1674
MSC: Primary 11R23; Secondary 11Y40
MathSciNet review: 1308453
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Abstract: For distinct primes l and p, the Iwasawa invariant $ \lambda _l^ - $ stabilizes in the cyclotomic $ {\mathbb{Z}_p}$-tower over a complex abelian base field. We calculate this stable invariant for $ p = 3$ and various l and K. Our motivation was to search for a formula of Riemann-Hurwitz type for $ \lambda _l^ - $ that might hold in a p-extension. From our numerical results, it is clear that no formula of such a simple kind can hold. In the course of our calculations, we develop a method of computing $ \lambda _l^ - $ for an arbitrary complex abelian field and, for $ p = 3$, we make effective Washington's theorem on the stabilization of the l-part of the class group in the cyclotomic $ {\mathbb{Z}_p}$-extension. A new proof of this theorem is given in the appendix.

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Keywords: Relative class number, $ {\mathbb{Z}_l}$-extension, Iwasawa lambda invariant
Article copyright: © Copyright 1995 American Mathematical Society

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