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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, <!– MATH ${\mathbb {Z}_l}$ –> <IMG WIDTH="25" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${\mathbb {Z}_l}$">-extension, Iwasawa lambda invariant
Article copyright: © Copyright 1995 American Mathematical Society