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

### Volume 94 / 2000

Acta Arithmetica 94 (2000), 365-371 DOI: 10.4064/aa-94-4-365-371

#### Abstract

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

• Gen Yamamoto

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