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On Minkowski units constructed by special values of Siegel modular functions

Takashi Fukuda;  Keiichi Komatsu — 2003

Journal de théorie des nombres de Bordeaux

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 ≥ 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 10 7 . 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 · 10 8 . Our argument also leads to the conclusion that h n ( n ≥ 1 ) is not divisible by a prime number...

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