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Using the special values of Siegel modular functions, we construct Minkowski units for the ray class field $k_{6}$ of $ℚ (e x p (2 π i / 5))$ modulo $6$ . Our work is based on investigating the prime decomposition of the special values and describing explicitly the action of the Galois group $G (k_{6} / ℚ)$ for the special values. Futhermore we construct the full unit group of $k_{6}$ using modular and circular units under the GRH.

Weber’s class number problem in the cyclotomic $ℤ_{2}$ -extension of $ℚ$ , II

Takashi Fukuda; Keiichi Komatsu — 2010

Journal de Théorie des Nombres de Bordeaux

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...

Iwasawa λ-invariants and Mordell-Weil ranks of abelian varieties with complex multiplication

Takashi Fukuda; Keiichi Komatsu; Shuji Yamagata — 2007

Acta Arithmetica

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Noncyclotomic $ℤ_{p}$ -extensions of imaginary quadratic fields.

Normal basis and Greenberg' s conjecture.

K-groups and ideal class groups of number fields.

On adele rings of arithmetically equivalent fields

On the Iwasawa λ-invariants of quaternion extensions

On Minkowski units constructed by special values of Siegel modular functions

Weber’s class number problem in the cyclotomic $ℤ_{2}$ -extension of $ℚ$ , II

Iwasawa λ-invariants and Mordell-Weil ranks of abelian varieties with complex multiplication