## Journal of the Mathematical Society of Japan

### On a bound of $\lambda$ and the vanishing of $\mu$ of $\mathbb{Z}_p$-extensions of an imaginary quadratic field

Satoshi FUJII

#### Abstract

Let $p$ be an odd prime number. To ask the behavior of $\lambda$- and $\mu$-invariants is a basic problem in Iwasawa theory of $\mathbb{Z}_p$-extensions. Sands showed that if $p$ does not divide the class number of an imaginary quadratic field $k$ and if the $\lambda$-invariant of the cyclotomic $\mathbb{Z}_p$-extension of $k$ is 2, then $\mu$-invariants vanish for all $\mathbb{Z}_p$-extensions of $k$, and $\lambda$-invariants are less than or equal to 2 for $\mathbb{Z}_p$-extensions of $k$ in which all primes above $p$ are totally ramified. In this article, we show results similar to Sands' results without the assumption that $p$ does not divide the class number of $k$. When $\mu$-invariants vanish, we also give an explicit upper bound of $\lambda$-invariants of all $\mathbb{Z}_p$-extensions.

#### Article information

Source
J. Math. Soc. Japan, Volume 65, Number 1 (2013), 277-298.

Dates
First available in Project Euclid: 24 January 2013

https://projecteuclid.org/euclid.jmsj/1359036455

Digital Object Identifier
doi:10.2969/jmsj/06510277

Mathematical Reviews number (MathSciNet)
MR3034405

Zentralblatt MATH identifier
1275.11141

Subjects
Primary: 11R23: Iwasawa theory

#### Citation

FUJII, Satoshi. On a bound of $\lambda$ and the vanishing of $\mu$ of $\mathbb{Z}_p$-extensions of an imaginary quadratic field. J. Math. Soc. Japan 65 (2013), no. 1, 277--298. doi:10.2969/jmsj/06510277. https://projecteuclid.org/euclid.jmsj/1359036455

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