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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On small Iwasawa invariants and imaginary quadratic fields
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by Jonathan W. Sands PDF
Proc. Amer. Math. Soc. 112 (1991), 671-684 Request permission

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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Additional Information
  • © Copyright 1991 American Mathematical Society
  • 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