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

Acta Arithmetica (2000)

• Volume: 94, Issue: 4, page 365-371
• ISSN: 0065-1036

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## Abstract

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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, ${k}_{\infty }$ a ${ℤ}_{p}$-extension of k, ${k}_{n}$ the nth layer of ${k}_{\infty }/k$, 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 ${k}_{\infty }/k$ 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 ${k}_{\infty }/k$. Then there exist integers $\lambda =\lambda \left({k}_{\infty }/k\right)\ge 0$, $\mu =\mu \left({k}_{\infty }/k\right)\ge 0$, $\nu =\nu \left({k}_{\infty }/k\right)$, and n₀ ≥ 0 such that ${A}_{n}={p}^{\lambda n+\mu {p}^{n}+\nu }$ for all n ≥ n₀, where ${A}_{n}$ is the order of ${A}_{n}$. These integers $\lambda =\lambda \left({k}_{\infty }/k\right)$, $\mu =\mu \left({k}_{\infty }/k\right)$ and $\nu =\nu \left({k}_{\infty }/k\right)$ are called Iwasawa invariants of ${k}_{\infty }/k$ for p. If ${k}_{\infty }$ is the cyclotomic ${ℤ}_{p}$-extension of k, then we denote λ (resp. μ and ν) by ${\lambda }_{p}\left(k\right)$ (resp. ${\mu }_{p}\left(k\right)$ and ${\nu }_{p}\left(k\right)$). Ferrero and Washington proved ${\mu }_{p}\left(k\right)=0$ for any abelian extension field k of ℚ. On the other hand, Greenberg [4] conjectured that if k is a totally real, then ${\lambda }_{p}\left(k\right)={\mu }_{p}\left(k\right)=0$. We call this conjecture Greenberg’s conjecture. In this paper, we determine all absolutely abelian p-extensions k with ${\lambda }_{p}\left(k\right)={\mu }_{p}\left(k\right)={\nu }_{p}\left(k\right)=0$ for an odd prime p, by using the results of G. Cornell and M. Rosen [1].

## How to cite

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Gen Yamamoto. "On the vanishing of Iwasawa invariants of absolutely abelian p-extensions." Acta Arithmetica 94.4 (2000): 365-371. <http://eudml.org/doc/207435>.

@article{GenYamamoto2000,
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].},
author = {Gen Yamamoto},
journal = {Acta Arithmetica},
keywords = {Iwasawa invariants; cyclic p-extensions; class group; genus field},
language = {eng},
number = {4},
pages = {365-371},
title = {On the vanishing of Iwasawa invariants of absolutely abelian p-extensions},
url = {http://eudml.org/doc/207435},
volume = {94},
year = {2000},
}

TY - JOUR
AU - Gen Yamamoto
TI - On the vanishing of Iwasawa invariants of absolutely abelian p-extensions
JO - Acta Arithmetica
PY - 2000
VL - 94
IS - 4
SP - 365
EP - 371
AB - 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].
LA - eng
KW - Iwasawa invariants; cyclic p-extensions; class group; genus field
UR - http://eudml.org/doc/207435
ER -

## References

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1. [1] G. Cornell and M. Rosen, The class group of an absolutely abelian l-extension, Illinois J. Math. 32 (1988), 453-461. Zbl0654.12006
2. [2] T. Fukuda, On the vanishing of Iwasawa invariants of certain cyclic extensions of ℚ with prime degree, Proc. Japan Acad. 73 (1997), 108-110. Zbl0899.11052
3. [3] T. Fukuda, K. Komatsu, M. Ozaki and H. Taya, On Iwasawa ${\lambda }_{p}$-invariants of relative real cyclic extension of degree p, Tokyo J. Math. 20 (1997), 475-480. Zbl0919.11068
4. [4] R. Greenberg, On the Iwasawa invariants of totally real number fields, Amer. J. Math. 98 (1976), 263-284. Zbl0334.12013
5. [5] K. Iwasawa, A note on class numbers of algebraic number fields, Abh. Math. Sem. Univ. Hamburg 20 (1956), 257-258. Zbl0074.03002
6. [6] G. Yamamoto, On the vanishing of Iwasawa invariants of certain (p,p)-extensions of ℚ, Proc. Japan Acad. 73A (1997), 45-47. Zbl0879.11061

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