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

Fukuda, Takashi; Komatsu, Keiichi

Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 2, pp. 359-368

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 $\ell \equiv 3,5(mod8)$. 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 $1.2·{10}^8$. Our argument also leads to the conclusion that $h_n(n\ge 1)$ is not divisible by a prime number $\ell$ satisfying $\ell \neg \equiv \pm 1(mod16)$.

Soit $h_n$ le nombres de classes du $n$-ième étage de la ${\mathbb{Z}}_2$-extension cyclotomique de $\mathbb{Q}$. Weber a prouvé que $h_n(n\ge 1)$ est impair et Horie a prouvé que $h_n(n\ge 1)$ n’est divisible par aucun nombre premier $\ell$ satisfaisant $\ell \equiv 3,5(mod8)$. Dans un article précédent, les auteurs ont montré $h_n(n\ge 1)$ n’est divisible par aucun nombre premier $\ell$ inférieur à ${10}^7$. Dans le présent article, en étudiant plus précisément les propriétés d’une unité particulière, nous montrons que $h_n(n\ge 1)$ n’est divisible par aucun nombre premier $\ell$ inférieur à $1.2·{10}^8$. Notre argument conduit aussi à la conclusion que $h_n(n\ge 1)$ n’est divisible par aucun nombre premier $\ell$ satisfaisant $\ell \neg \equiv \pm 1(mod16)$.

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