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Elliptic unit groups. (Groupes d’unités elliptiques.) (French) Zbl 0434.12003
Let \(H\) be any finite real abelian extension field of the rational number field \(\mathbb Q\), and denote by \(E_H\), \(h_H\) and \(G=\mathrm{Gal}(H/\mathbb Q)\) the unit group, the ideal class number of \(H\) and the Galois group of \(H/\mathbb Q\) respectively. Then, H. W. Leopoldt constructed a group \(C_H\) of cyclotomic units of \(H\) and showed that the group index \([E_H:C_H]\) is the product of the class number \(h_H\) by an integer \(Q_G\) which depends only on the structure of \(G\) [Abh. Deutsch. Akad. Wiss. Berlin, Math.-Naturw. Kl. 1953, No. 2, 1–48 (1954; Zbl 0059.03501)] .
In this paper, the authors give a generalization of Leopoldt’s result. Namely, let \(H/K\) be a finite abelian extension of an imaginary quadratic number field \(K\), \(H_{(1)}\) be the Hilbert class field and \(H_0 =H\cap H_{(1)}\) be the maximal unramified extension field of \(K\) in \(H\). Denote by \(G\) and \(\mathfrak G\) the Galois group of \(H/H_0\) and \(H/K\), respectively. Then, by a similar method to the Leopoldt’s one, they first construct a group of elliptic units \(\Theta_{I(\mathfrak G)}\) which is \(Z[\mathfrak G]\)-isomorphic to the augmentation ideal \(I(\mathfrak G)\) of the group algebra \(Z[\mathfrak G]\). Next, they construct two groups \(V_3\), \(V_4\) of elliptic units in \(H\) such that \[ \Theta_{I(\mathfrak G)} \subset V_3\subset V_4\subset E_H.\]
Finally, they show that the group indices \([E_H:V_3]\) and \([E_H:V_4]\) are related to the ideal class number \(h_H\) of \(H\) by the following formula analogous to the Leopoldt’s one: \[ [E_H:V_3] =c_3\cdot h_H/[H_{(1)}:H_0],\quad [E_H:V_4] = c_4\cdot h_H/h_{H_0}\] where \(c_3\) and \(c_4\) are integers both of which depend only on the structure of \(G=\mathrm{Gal}(H/H_0)\) and the degree \([H_0:K]\) of the extension \(H_0/K\).
Reviewer: Hideo Yokoi

MSC:
11G16 Elliptic and modular units
11R37 Class field theory
11R29 Class numbers, class groups, discriminants
11R18 Cyclotomic extensions
11R11 Quadratic extensions
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