On the vanishing of Iwasawa invariants of absolutely abelian p-extensions

1. Introduction. Let p be a prime number and ${\displaystyle {\mathbb{Z}}_p}$ the ring of p-adic integers. Let k be a finite extension of the rational number field ℚ, ${\displaystyle k_{\infty }}$ a ${\displaystyle {\mathbb{Z}}_p}$-extension of k, ${\displaystyle k_n}$ the nth layer of ${\displaystyle \frac{k_{\infty }}{k}}$, and ${\displaystyle A_n}$ the p-Sylow subgroup of the ideal class group of ${\displaystyle k_n}$. Iwasawa proved the following well-known theorem about the order ${\displaystyle \#A_n}$ of ${\displaystyle A_n}$: Theorem A (Iwasawa). Let ${\displaystyle \frac{k_{\infty }}{k}}$ be a ${\displaystyle {\mathbb{Z}}_p}$-extension and ${\displaystyle A_n}$ the p-Sylow subgroup of the ideal class group of ${\displaystyle k_n}$, where ${\displaystyle k_n}$ is the ${\displaystyle n}$th layer of ${\displaystyle \frac{k_{\infty }}{k}}$. Then there exist integers ${\displaystyle \lambda =\lambda \left(\frac{k_{\infty }}{k}\right)\ge 0}$, ${\displaystyle \mu =\mu \left(\frac{k_{\infty }}{k}\right)\ge 0}$, ${\displaystyle \nu =\nu \left(\frac{k_{\infty }}{k}\right)}$, and n₀ ≥ 0 such that ${\displaystyle \#A_n=p^{\lambda n+\mu p^n+\nu }}$ for all n ≥ n₀, where ${\displaystyle \#A_n}$ is the order of ${\displaystyle A_n}$. These integers ${\displaystyle \lambda =\lambda \left(\frac{k_{\infty }}{k}\right)}$, ${\displaystyle \mu =\mu \left(\frac{k_{\infty }}{k}\right)}$ and ${\displaystyle \nu =\nu \left(\frac{k_{\infty }}{k}\right)}$ are called Iwasawa invariants of ${\displaystyle \frac{k_{\infty }}{k}}$ for p. If ${\displaystyle k_{\infty }}$ is the cyclotomic ${\displaystyle {\mathbb{Z}}_p}$-extension of k, then we denote λ (resp. μ and ν) by ${\displaystyle {\lambda }_p\left(k\right)}$ (resp. ${\displaystyle {\mu }_p\left(k\right)}$ and ${\displaystyle {\nu }_p\left(k\right)}$). Ferrero and Washington proved ${\displaystyle {\mu }_p\left(k\right)=0}$ for any abelian extension field k of ℚ. On the other hand, Greenberg [4] conjectured that if k is a totally real, then ${\displaystyle {\lambda }_p\left(k\right)={\mu }_p\left(k\right)=0}$. We call this conjecture Greenberg's conjecture. In this paper, we determine all absolutely abelian p-extensions k with ${\displaystyle {\lambda }_p\left(k\right)={\mu }_p\left(k\right)={\nu }_p\left(k\right)=0}$ for an odd prime p, by using the results of G. Cornell and M. Rosen [1].