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On Minkowski units constructed by special values of Siegel modular functions

Takashi FukudaKeiichi 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 ( 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 / ) for the special values. Futhermore we construct the full unit group of k 6 using modular and circular units under the GRH.

The Iwasawa λ-invariants of ℤₚ-extensions of real quadratic fields

Takashi FukudaHisao Taya — 1995

Acta Arithmetica

1. Introduction. Let k be a totally real number field. Let p be a fixed prime number and ℤₚ the ring of all p-adic integers. We denote by λ=λₚ(k), μ=μₚ(k) and ν=νₚ(k) the Iwasawa invariants of the cyclotomic ℤₚ-extension k of k for p (cf. [10]). Then Greenberg’s conjecture states that both λₚ(k) and μₚ(k) always vanish (cf. [8]). In other words, the order of the p-primary part of the ideal class group of kₙ remains bounded as n tends to infinity, where kₙ is the nth layer of k / k . We know by the Ferrero-Washington...

Weber’s class number problem in the cyclotomic 2 -extension of , II

Takashi FukudaKeiichi Komatsu — 2010

Journal de Théorie des Nombres de Bordeaux

Let h n denote the class number of n -th layer of the cyclotomic 2 -extension of . Weber proved that h n ( n 1 ) is odd and Horie proved that h n ( n 1 ) is not divisible by a prime number satisfying 3 , 5 ( mod 8 ) . In a previous paper, the authors showed that h n ( n 1 ) 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 ( n 1 ) is not divisible by a prime number 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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