×

zbMATH — the first resource for mathematics

On the cyclotomic unit group and the ideal class group of a real abelian number field. (English) Zbl 0879.11058
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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ferrero, B.; Washington, L.C., The Iwasawa invariantμpvanishes for abelian number fields, Ann. math., 109, 377-395, (1979) · Zbl 0443.12001
[2] Greenberg, R., On the Iwasawa invariants of totally real number fields, Amer. J. math., 98, 263-284, (1976) · Zbl 0334.12013
[3] Greither, C., Class groups of abelian fields, and the main conjecture, Ann. inst. Fourier, 42, 449-499, (1992) · Zbl 0729.11053
[4] Greither, C., Über relative-invariante kreiseinheiten und Stickelberger-elemente, Manuscripta math., 80, 27-43, (1993) · Zbl 0801.11044
[5] Iwasawa, K., Onγ, Bull. amer. math. soc., 65, 183-226, (1959) · Zbl 0089.02402
[6] Iwasawa, K., Some problems on cyclotomic fields, RIMS kokyuroku, 658, 43-55, (1988)
[7] Janusz, G.J., Algebraic number fields, (1973), Academic Press New York · Zbl 0307.12001
[8] Kim, J.M., Cohomology groups of cyclotomic units, J. algebra, 152, 514-519, (1992) · Zbl 0776.11066
[9] Kraft, J.; Schoof, R., Compositio math., 97, 135-155, (1995)
[10] Mazur, B.; Wiles, A., Class fields of abelian extensions of Q, Invent. math., 76, 179-330, (1984) · Zbl 0545.12005
[11] Rubin, K., The main conjecture, ()
[12] Sinnott, W., On the Stickelberger ideal and the circular units of an abelian field, Invent. math., 62, 181-234, (1980) · Zbl 0465.12001
[13] Washington, L.C., Introduction to cyclotomic fields, Graduate texts in mathematics, 83, (1982), Springer New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.