# 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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topGen 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

top- [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] 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] 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] R. Greenberg, On the Iwasawa invariants of totally real number fields, Amer. J. Math. 98 (1976), 263-284. Zbl0334.12013
- [5] K. Iwasawa, A note on class numbers of algebraic number fields, Abh. Math. Sem. Univ. Hamburg 20 (1956), 257-258. Zbl0074.03002
- [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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