The Iwasawa λ-invariants of ℤₚ-extensions of real quadratic fields

Takashi Fukuda; Hisao Taya

Acta Arithmetica (1995)

  • Volume: 69, Issue: 3, page 277-292
  • ISSN: 0065-1036

Abstract

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1. Introduction. Let k be a totally real number field. Let p be a fixed prime number and ℤₚ the ring of all p-adic integers. We denote by λ=λₚ(k), μ=μₚ(k) and ν=νₚ(k) the Iwasawa invariants of the cyclotomic ℤₚ-extension k ∞ of k for p (cf. [10]). Then Greenberg’s conjecture states that both λₚ(k) and μₚ(k) always vanish (cf. [8]). In other words, the order of the p-primary part of the ideal class group of kₙ remains bounded as n tends to infinity, where kₙ is the nth layer of k ∞ / k . We know by the Ferrero-Washington theorem (cf. [2], [15]) that μₚ(k) always vanishes when k is an abelian (not necessarily totally real) number field. However, the conjecture remains unsolved up to now except for some special cases (cf. [1], [3], [5]-[8], [13]). This paper is a continuation of our previous papers [3], [5]-[7] and [12], that is to say, we investigate Greenberg’s conjecture when k is a real quadratic field and p is an odd prime number which splits in k. The purpose of this paper is to extend our previous results, and to give basic numerical data of k=ℚ(√m) for 0 ≤ m ≤ 10000 and p=3. On the basis of these data, we can verify Greenberg’s conjecture for most of these k’s.

How to cite

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Takashi Fukuda, and Hisao Taya. "The Iwasawa λ-invariants of ℤₚ-extensions of real quadratic fields." Acta Arithmetica 69.3 (1995): 277-292. <http://eudml.org/doc/206688>.

@article{TakashiFukuda1995,
abstract = {1. Introduction. Let k be a totally real number field. Let p be a fixed prime number and ℤₚ the ring of all p-adic integers. We denote by λ=λₚ(k), μ=μₚ(k) and ν=νₚ(k) the Iwasawa invariants of the cyclotomic ℤₚ-extension $k_∞$ of k for p (cf. [10]). Then Greenberg’s conjecture states that both λₚ(k) and μₚ(k) always vanish (cf. [8]). In other words, the order of the p-primary part of the ideal class group of kₙ remains bounded as n tends to infinity, where kₙ is the nth layer of $k_∞/k$. We know by the Ferrero-Washington theorem (cf. [2], [15]) that μₚ(k) always vanishes when k is an abelian (not necessarily totally real) number field. However, the conjecture remains unsolved up to now except for some special cases (cf. [1], [3], [5]-[8], [13]). This paper is a continuation of our previous papers [3], [5]-[7] and [12], that is to say, we investigate Greenberg’s conjecture when k is a real quadratic field and p is an odd prime number which splits in k. The purpose of this paper is to extend our previous results, and to give basic numerical data of k=ℚ(√m) for 0 ≤ m ≤ 10000 and p=3. On the basis of these data, we can verify Greenberg’s conjecture for most of these k’s.},
author = {Takashi Fukuda, Hisao Taya},
journal = {Acta Arithmetica},
keywords = {Greenberg's conjecture; Iwasawa invariants; totally real number field; real quadratic field; table},
language = {eng},
number = {3},
pages = {277-292},
title = {The Iwasawa λ-invariants of ℤₚ-extensions of real quadratic fields},
url = {http://eudml.org/doc/206688},
volume = {69},
year = {1995},
}

TY - JOUR
AU - Takashi Fukuda
AU - Hisao Taya
TI - The Iwasawa λ-invariants of ℤₚ-extensions of real quadratic fields
JO - Acta Arithmetica
PY - 1995
VL - 69
IS - 3
SP - 277
EP - 292
AB - 1. Introduction. Let k be a totally real number field. Let p be a fixed prime number and ℤₚ the ring of all p-adic integers. We denote by λ=λₚ(k), μ=μₚ(k) and ν=νₚ(k) the Iwasawa invariants of the cyclotomic ℤₚ-extension $k_∞$ of k for p (cf. [10]). Then Greenberg’s conjecture states that both λₚ(k) and μₚ(k) always vanish (cf. [8]). In other words, the order of the p-primary part of the ideal class group of kₙ remains bounded as n tends to infinity, where kₙ is the nth layer of $k_∞/k$. We know by the Ferrero-Washington theorem (cf. [2], [15]) that μₚ(k) always vanishes when k is an abelian (not necessarily totally real) number field. However, the conjecture remains unsolved up to now except for some special cases (cf. [1], [3], [5]-[8], [13]). This paper is a continuation of our previous papers [3], [5]-[7] and [12], that is to say, we investigate Greenberg’s conjecture when k is a real quadratic field and p is an odd prime number which splits in k. The purpose of this paper is to extend our previous results, and to give basic numerical data of k=ℚ(√m) for 0 ≤ m ≤ 10000 and p=3. On the basis of these data, we can verify Greenberg’s conjecture for most of these k’s.
LA - eng
KW - Greenberg's conjecture; Iwasawa invariants; totally real number field; real quadratic field; table
UR - http://eudml.org/doc/206688
ER -

References

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