Weber’s class number problem in the cyclotomic ℤ 2 -extension of ℚ , II

Takashi Fukuda[1]; Keiichi Komatsu[2]

  • [1] Department of Mathematics College of Industrial Technology Nihon University 2-11-1 Shin-ei, Narashino, Chiba, Japan
  • [2] Department of Mathematics School of Science and Engineering Waseda University 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan

Journal de Théorie des Nombres de Bordeaux (2010)

  • Volume: 22, Issue: 2, page 359-368
  • ISSN: 1246-7405

Abstract

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Let h n denote the class number of n -th layer of the cyclotomic ℤ 2 -extension of ℚ . Weber proved that h n ( n ≥ 1 ) is odd and Horie proved that h n ( n ≥ 1 ) is not divisible by a prime number ℓ satisfying ℓ ≡ 3 , 5 ( mod 8 ) . In a previous paper, the authors showed that h n ( n ≥ 1 ) is not divisible by a prime number ℓ less than 10 7 . In this paper, by investigating properties of a special unit more precisely, we show that h n ( n ≥ 1 ) is not divisible by a prime number ℓ less than 1 . 2 · 10 8 . Our argument also leads to the conclusion that h n ( n ≥ 1 ) is not divisible by a prime number ℓ satisfying ℓ ¬ ≡ ± 1 ( mod 16 ) .

How to cite

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Fukuda, Takashi, and Komatsu, Keiichi. "Weber’s class number problem in the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}$, II." Journal de Théorie des Nombres de Bordeaux 22.2 (2010): 359-368. <http://eudml.org/doc/116408>.

@article{Fukuda2010,
abstract = {Let $h_n$ denote the class number of $n$-th layer of the cyclotomic $\mathbb\{Z\}_2$-extension of $\mathbb\{Q\}$. Weber proved that $h_n\;\;(n\ge 1)$ is odd and Horie proved that $h_n\;\;(n\ge 1)$ is not divisible by a prime number $\ell $ satisfying $\ell \equiv 3,\,5\hspace\{4.44443pt\}(\@mod \; 8)$. In a previous paper, the authors showed that $h_n\;\;(n\ge 1)$ is not divisible by a prime number $\ell $ less than $10^7$. In this paper, by investigating properties of a special unit more precisely, we show that $h_n\;\;(n\ge 1)$ is not divisible by a prime number $\ell $ less than $1.2\cdot 10^8$. Our argument also leads to the conclusion that $h_n\;\;(n\ge 1)$ is not divisible by a prime number $\ell $ satisfying $\ell \lnot \equiv \pm \;1\hspace\{4.44443pt\}(\@mod \; 16)$.},
affiliation = {Department of Mathematics College of Industrial Technology Nihon University 2-11-1 Shin-ei, Narashino, Chiba, Japan; Department of Mathematics School of Science and Engineering Waseda University 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan},
author = {Fukuda, Takashi, Komatsu, Keiichi},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {cyclotomic -extension of ; class number},
language = {eng},
number = {2},
pages = {359-368},
publisher = {Université Bordeaux 1},
title = {Weber’s class number problem in the cyclotomic $\mathbb\{Z\}_2$-extension of $\mathbb\{Q\}$, II},
url = {http://eudml.org/doc/116408},
volume = {22},
year = {2010},
}

TY - JOUR
AU - Fukuda, Takashi
AU - Komatsu, Keiichi
TI - Weber’s class number problem in the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}$, II
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2010
PB - Université Bordeaux 1
VL - 22
IS - 2
SP - 359
EP - 368
AB - Let $h_n$ denote the class number of $n$-th layer of the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}$. Weber proved that $h_n\;\;(n\ge 1)$ is odd and Horie proved that $h_n\;\;(n\ge 1)$ is not divisible by a prime number $\ell $ satisfying $\ell \equiv 3,\,5\hspace{4.44443pt}(\@mod \; 8)$. In a previous paper, the authors showed that $h_n\;\;(n\ge 1)$ is not divisible by a prime number $\ell $ less than $10^7$. In this paper, by investigating properties of a special unit more precisely, we show that $h_n\;\;(n\ge 1)$ is not divisible by a prime number $\ell $ less than $1.2\cdot 10^8$. Our argument also leads to the conclusion that $h_n\;\;(n\ge 1)$ is not divisible by a prime number $\ell $ satisfying $\ell \lnot \equiv \pm \;1\hspace{4.44443pt}(\@mod \; 16)$.
LA - eng
KW - cyclotomic -extension of ; class number
UR - http://eudml.org/doc/116408
ER -

References

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  1. H. Bauer, Numeriche Bestimmung von Klassenzahlen reeller zyklischer Zahlkörper. J. Number Theory 1 (1969), 161–162. Zbl0167.32301MR240072
  2. H. Cohn, A numerical study of Weber’s real class number calculation I. Numer. Math. 2 (1960), 347–362. Zbl0117.27501MR122809
  3. T. Fukuda and K. Komatsu, Weber’s class number problem in the cyclotomic ℤ 2 -extension of ℚ . Experimental Math. 18 (2009), 213–222. Zbl1189.11033MR2549691
  4. K. Horie, Ideal class groups of Iwasawa-theoritical abelian extensions over the rational field. J. London Math. Soc. 66 (2002), 257–275. Zbl1011.11072MR1920401
  5. K. Horie, The ideal class group of the basic ℤ p -extension over an imaginary quadratic field. Tohoku Math. J. 57 (2005), 375–394. Zbl1128.11051MR2154097
  6. K. Horie, Triviality in ideal class groups of Iwasawa-theoretical abelian number fields. J. Math. Soc. Japan 57 (2005), 827–857. Zbl1160.11357MR2139736
  7. K. Horie, Primary components of the ideal class groups of Iwasawa-theoretical abelian number field. J. Math. Soc. Japan 59 (2007), 811–824. Zbl1128.11052MR2344829
  8. K. Horie, Certain primary components of the ideal class group of the ℤ p -extension over the rationals. Tohoku Math. J. 59 (2007), 259–291. Zbl1202.11050MR2347423
  9. K. Horie and M. Horie, The ideal class group of the ℤ p -extension over the rationals. Tohoku Math. J., 61 (2009), 551–570. Zbl1238.11101MR2598249
  10. J. M. Masley, Class numbers of real cyclic number fields with small conductor. Compositio Math. 37 (1978), 297–319. Zbl0428.12003MR511747
  11. F. J. van der Linden, Class Number Computations of Real Abelian Number Fields. Math. Comp. 39 (1982), 693–707. Zbl0505.12010MR669662
  12. L. C. Washington, Class numbers and ℤ p -extensions. Math. Ann. 214 (1975), 177–193. Zbl0302.12007MR364182
  13. L. C. Washington, The non- p -part of the class number in a cyclotomic ℤ p -extension. Inv. Math. 49 (1978), 87–97. Zbl0403.12007MR511097
  14. H. Weber, Theorie der Abel’schen Zahlkörper. Acta Math. 8 (1886), 193–263. MR1554698

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