Weber’s class number problem in the cyclotomic ℤ2-extension of ℚ, II

Takashi Fukuda; Keiichi Komatsu

doi:10.5802/jtnb.720

Weber’s class number problem in the cyclotomic ℤ₂-extension of ℚ, II

Takashi Fukuda, Keiichi Komatsu

Research output: Contribution to journal › Article › peer-review

9 Citations (Scopus)

Abstract

Let h_n denote the class number of n-th layer of the cyclotomic ℤ₂-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 10⁷. 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 • 10⁸. 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).

Original language	English
Pages (from-to)	359-368
Number of pages	10
Journal	Journal de Theorie des Nombres de Bordeaux
Volume	22
Issue number	2
DOIs	https://doi.org/10.5802/jtnb.720
Publication status	Published - 2010

ASJC Scopus subject areas

Algebra and Number Theory

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title = "Weber{\textquoteright}s class number problem in the cyclotomic ℤ2-extension of ℚ, II",

abstract = "Let hn denote the class number of n-th layer of the cyclotomic ℤ2-extension of ℚ. Weber proved that hn (n ≥ 1) is odd and Horie proved that hn (n ≥ 1) is not divisible by a prime number ℓ satisfying ℓ ≡ 3, 5 (mod 8). In a previous paper, the authors showed that hn (n ≥ 1) is not divisible by a prime number ℓ less than 107. In this paper, by investigating properties of a special unit more precisely, we show that hn (n ≥ 1) is not divisible by a prime number ℓ less than 1.2 • 108. Our argument also leads to the conclusion that hn (n ≥ 1) is not divisible by a prime number ℓ satisfying ℓ = ± 1 (mod 16).",

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Weber’s class number problem in the cyclotomic ℤ₂-extension of ℚ, II

Abstract

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