On the cyclotomic unit group and the ideal class group of a real abelian number field.

*(English)*Zbl 0879.11058Let \(p\) 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.

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
Full Text:
DOI

##### References:

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