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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  1. [1] A. Candiotti, Computations of Iwasawa invariants and K₂, Compositio Math. 29 (1974), 89-111. Zbl0364.12003
  2. [2] B. Ferrero and L. C. Washington, The Iwasawa invariant μₚ vanishes for abelian number fields, Ann. of Math. 109 (1979), 377-395. Zbl0443.12001
  3. [3] T. Fukuda, Iwasawa λ-invariants of certain real quadratic fields, Proc. Japan Acad. 65A (1989), 260-262. Zbl0703.11055
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  8. [8] R. Greenberg, On the Iwasawa invariants of totally real number fields, Amer. J. Math. 98 (1976), 263-284. Zbl0334.12013
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  10. [10] K. Iwasawa, On ℤ l -extensions of algebraic number fields, Ann. of Math. 98 (1973), 246-326. Zbl0285.12008
  11. [11] S. Mäki, The determination of units in real cyclic sextic fields, Lecture Notes in Math. 797, Springer, Berlin, 1980. Zbl0423.12006
  12. [12] H. Taya, On the Iwasawa λ-invariants of real quadratic fields, Tokyo J. Math. 16 (1993), 121-130. Zbl0797.11084
  13. [13] H. Taya, Computation of ℤ₃-invariants of real quadratic fields, preprint series, Waseda Univ. Technical Report No. 93-13, 1993. 
  14. [14] H. Wada and M. Saito, A table of ideal class groups of imaginary quadratic fields, Sophia Kôkyuroku in Math. 28, Depart. of Math., Sophia Univ. Tokyo, 1988. Zbl0629.12003
  15. [15] L. C. Washington, Introduction to Cyclotomic Fields, Graduate Texts in Math. 83, Springer, New York, 1982. 
  16. [16] H. Yokoi, On the class number of a relatively cyclic number field, Nagoya Math. J. 29 (1967), 31-44. Zbl0166.05803

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