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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On small Iwasawa invariants and imaginary quadratic fields


Author: Jonathan W. Sands
Journal: Proc. Amer. Math. Soc. 112 (1991), 671-684
MSC: Primary 11R23
DOI: https://doi.org/10.1090/S0002-9939-1991-1057961-4
MathSciNet review: 1057961
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Abstract: If $p$ is an odd prime that does not divide the class number of the imaginary quadratic field $k$, and the cyclotomic ${\mathbb {Z}_p}$-extension of $k$ has $\lambda$-invariant less than or equal to two, we prove that every totally ramified ${\mathbb {Z}_p}$-extension of $k$ has $\mu$-invariant equal to zero and $\lambda$-invariant less than or equal to two. Combined with a result of Bloom and Gerth, this has the consequence that $\mu = 0$ for every ${\mathbb {Z}_p}$-extension of $k$, under the same assumptions. In the principal case under consideration, Iwasawa’s formula for the power of $p$ in the class number of the $n$th layer of a ${\mathbb {Z}_p}$-extension becomes valid for all $n$ , and is completely explicit.


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Keywords: Iwasawa invariant, distinguished polynomial, class field
Article copyright: © Copyright 1991 American Mathematical Society