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

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.