Weber’s class number problem in the cyclotomic 2 -extension of , II
Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 2, pp. 359-368.

Soit h n le nombres de classes du n-ième étage de la 2 -extension cyclotomique de . Weber a prouvé que h n (n1) est impair et Horie a prouvé que h n (n1) n’est divisible par aucun nombre premier satisfaisant 3,5(mod8). Dans un article précédent, les auteurs ont montré h n (n1) n’est divisible par aucun nombre premier inférieur à 10 7 . Dans le présent article, en étudiant plus précisément les propriétés d’une unité particulière, nous montrons que h n (n1) n’est divisible par aucun nombre premier inférieur à 1.2·10 8 . Notre argument conduit aussi à la conclusion que h n (n1) n’est divisible par aucun nombre premier satisfaisant ¬±1(mod16).

Let h n denote the class number of n-th layer of the cyclotomic 2 -extension of . Weber proved that h n (n1) is odd and Horie proved that h n (n1) is not divisible by a prime number satisfying 3,5(mod8). In a previous paper, the authors showed that h n (n1) is not divisible by a prime number less than 10 7 . In this paper, by investigating properties of a special unit more precisely, we show that h n (n1) is not divisible by a prime number less than 1.2·10 8 . Our argument also leads to the conclusion that h n (n1) is not divisible by a prime number satisfying ¬±1(mod16).

DOI : 10.5802/jtnb.720
Fukuda, Takashi 1 ; Komatsu, Keiichi 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
@article{JTNB_2010__22_2_359_0,
     author = {Fukuda, Takashi and Komatsu, Keiichi},
     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},
     zbl = {1223.11133},
     mrnumber = {2769067},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jtnb.720/}
}
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
SP  - 359
EP  - 368
VL  - 22
IS  - 2
PB  - Université Bordeaux 1
UR  - http://archive.numdam.org/articles/10.5802/jtnb.720/
DO  - 10.5802/jtnb.720
LA  - en
ID  - JTNB_2010__22_2_359_0
ER  - 
%0 Journal Article
%A Fukuda, Takashi
%A Komatsu, Keiichi
%T Weber’s class number problem in the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}$, II
%J Journal de théorie des nombres de Bordeaux
%D 2010
%P 359-368
%V 22
%N 2
%I Université Bordeaux 1
%U http://archive.numdam.org/articles/10.5802/jtnb.720/
%R 10.5802/jtnb.720
%G en
%F JTNB_2010__22_2_359_0
Fukuda, Takashi; 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, Tome 22 (2010) no. 2, pp. 359-368. doi : 10.5802/jtnb.720. http://archive.numdam.org/articles/10.5802/jtnb.720/

[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 2 -extension of . 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 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 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 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 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 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

Cité par Sources :