On the cyclotomic unit group and the ideal class group of a real abelian number field. (English) Zbl 0879.11058

Ozaki, Manabu

J. Number Theory 64, No. 2, 211-222 (1997).

Let \(p\) be a fixed odd prime number and \(K\) a real abelian number field. For the cyclotomic \(\mathbb{Z}_p\)-extension of \(K\), denote by \(K_n\) its \(n\)th layer, by \(E_n\) the group of units in \(K_n\) and by \(C_n\) the group of cyclotomic units in \(K_n\) in the sense of W. Sinnott [Invent. Math. 62, 181-234 (1980; Zbl 0465.12001)]. Let \(A_n\) and \(B_n\) be the \(p\)-Sylow subgroups of the ideal class group of \(K_n\) and the quotient group \(E_n/C_n\), respectively. Then if \(K\) has degree prime to \(p\), then \(\# A_n= \#B_n\). Also, the Iwasawa main conjecture proved by B. Mazur and A. Wiles [Invent. Math. 76, 179-330 (1984; Zbl 0545.12005)] says that the characteristic ideal of the projective limit of \(A_n\) with respect to the norm maps is equal to that of \(B_n\).
In the paper under review, the author considers the naturally arising problem whether \(A_n\) is isomorphic to \(B_n\) as Galois modules for sufficiently large \(n\). Put \(\Gamma_{m,n}=\text{Gal}(K_m/K_n)\) and let \(\widehat H^i (\Gamma_{m,n}, M)\) be the \(i\)th Tate cohomology group for a \(\Gamma_{m,n}\)-module \(M\). He gives fantastic results on the relations between \(A_n\) and \(B_n\). The first result is as follows: Assume that \(K\) satisfies certain conditions which are cited as condition (C) in the paper under review. Then, as Galois modules, for all \(m\geq n\geq 0\), \(\widehat H^i (\Gamma_{m,n}, A_n) \simeq \widehat H^i (\Gamma_{m,n}, B_n)\) for \(i=0\), \(-1\), and \(\widehat H^i (\Gamma_{ m,n}, A_n)\) is isomorphic to the cokernel of the natural map \(A_n\to A_m^{\Gamma_{m,n}}\), or the kernel of the natural map \(A_n\to A_m\), according as \(i=0\) or \(-1\), where \(A_m^{\Gamma_{m,n}}\) is the \(\Gamma_{m,n}\)-invariant part of \(A_m\). Next, by using this, he shows that if we further assume that Greenberg’s conjecture is valid, then \(A_n \simeq B_n\) as Galois modules for sufficiently large \(n\). Here Greenberg’s conjecture states that the Iwasawa \(\lambda_p\)-invariant always vanishes for any real abelian number field [R. Greenberg, Am. J. Math. 98, 263-284 (1976; Zbl 0334.12013)].
Moreover, he studies sufficient conditions that a real abelian number field satisfies condition (C) and shows that if \(K\) has prime power conductor \(p^e\), or, if \(K\) has degree prime to \(p\) and \(p\) remains prime in \(K\), then \(K\) satisfies condition (C).
For Part II see Zbl 0879.11061 below.

Reviewer: H.Taya (Sendai)

MSC:

11R18	Cyclotomic extensions
11R27	Units and factorization
11R23	Iwasawa theory
11R37	Class field theory

Keywords:

cyclotomic \(\mathbb{Z}_ p\)-extension; \(p\)-Sylow subgroups of the ideal class group; Galois modules; real abelian number field; group of cyclotomic units

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