Weber’s class number problem in the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}$, II

Takashi Fukuda; Keiichi Komatsu

doi:10.5802/jtnb.720

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, Volume 22 (2010) no. 2, pp. 359-368.

Abstract
Résumé

Let $h_{n}$ denote the class number of $n$ -th layer of the cyclotomic $ℤ_{2}$ -extension of $ℚ$ . Weber proved that $h_{n} (n \geq 1)$ is odd and Horie proved that $h_{n} (n \geq 1)$ is not divisible by a prime number $ℓ$ satisfying $ℓ \equiv 3, 5 (mod 8)$ . In a previous paper, the authors showed that $h_{n} (n \geq 1)$ 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 $h_{n} (n \geq 1)$ is not divisible by a prime number $ℓ$ less than $1.2 \cdot 10^{8}$ . Our argument also leads to the conclusion that $h_{n} (n \geq 1)$ is not divisible by a prime number $ℓ$ satisfying $ℓ \neg \equiv \pm 1 (mod 16)$ .

Soit $h_{n}$ le nombres de classes du $n$ -ième étage de la $ℤ_{2}$ -extension cyclotomique de $ℚ$ . Weber a prouvé que $h_{n} (n \geq 1)$ est impair et Horie a prouvé que $h_{n} (n \geq 1)$ n’est divisible par aucun nombre premier $ℓ$ satisfaisant $ℓ \equiv 3, 5 (mod 8)$ . Dans un article précédent, les auteurs ont montré $h_{n} (n \geq 1)$ n’est divisible par aucun nombre premier $ℓ$ inférieur à $10^{7}$ . Dans le présent article, en étudiant plus précisément les propriétés d’une unité particulière, nous montrons que $h_{n} (n \geq 1)$ n’est divisible par aucun nombre premier $ℓ$ inférieur à $1.2 \cdot 10^{8}$ . Notre argument conduit aussi à la conclusion que $h_{n} (n \geq 1)$ n’est divisible par aucun nombre premier $ℓ$ satisfaisant $ℓ \neg \equiv \pm 1 (mod 16)$ .

MR Zbl

DOI: 10.5802/jtnb.720

Author's affiliations:

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

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     author = {Takashi Fukuda and Keiichi Komatsu},
     title = {Weber{\textquoteright}s class number problem in the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}$, {II}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {359--368},
     publisher = {Universit\'e Bordeaux 1},
     volume = {22},
     number = {2},
     year = {2010},
     doi = {10.5802/jtnb.720},
     mrnumber = {2769067},
     zbl = {1223.11133},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.720/}
}

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Takashi Fukuda; Keiichi Komatsu. Weber’s class number problem in the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}$, II. Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 2, pp. 359-368. doi : 10.5802/jtnb.720. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.720/

References
Cited by

[1] H. Bauer, Numeriche Bestimmung von Klassenzahlen reeller zyklischer Zahlkörper. J. Number Theory 1 (1969), 161–162. | MR | Zbl

[2] H. Cohn, A numerical study of Weber’s real class number calculation I. Numer. Math. 2 (1960), 347–362. | MR | Zbl

[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. | MR | Zbl

[4] K. Horie, Ideal class groups of Iwasawa-theoritical abelian extensions over the rational field. J. London Math. Soc. 66 (2002), 257–275. | MR | Zbl

[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. | MR | Zbl

[6] K. Horie, Triviality in ideal class groups of Iwasawa-theoretical abelian number fields. J. Math. Soc. Japan 57 (2005), 827–857. | MR | Zbl

[7] K. Horie, Primary components of the ideal class groups of Iwasawa-theoretical abelian number field. J. Math. Soc. Japan 59 (2007), 811–824. | MR | Zbl

[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. | MR

[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. | MR

[10] J. M. Masley, Class numbers of real cyclic number fields with small conductor. Compositio Math. 37 (1978), 297–319. | EuDML | Numdam | MR | Zbl

[11] F. J. van der Linden, Class Number Computations of Real Abelian Number Fields. Math. Comp. 39 (1982), 693–707. | MR | Zbl

[12] L. C. Washington, Class numbers and $ℤ_{p}$ -extensions. Math. Ann. 214 (1975), 177–193. | EuDML | MR | Zbl

[13] L. C. Washington, The non- $p$ -part of the class number in a cyclotomic $ℤ_{p}$ -extension. Inv. Math. 49 (1978), 87–97. | EuDML | MR | Zbl

[14] H. Weber, Theorie der Abel’schen Zahlkörper. Acta Math. 8 (1886), 193–263. | JFM | MR

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