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Using the special values of Siegel modular functions, we construct Minkowski units for the ray class field $k_6$ of $\mathbb{Q}\left(exp\right(2\pi i/5\left)\right)$ modulo $6$. Our work is based on investigating the prime decomposition of the special values and describing explicitly the action of the Galois group $G(k_6/\mathbb{Q})$ for the special values. Futhermore we construct the full unit group of $k_6$ using modular and circular units under the GRH.

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 $k_{\infty }$ 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 $k_{\infty }/k$. We know by the Ferrero-Washington...

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...