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Remarks on \(\mathbb{Z}_ p\)-extensions of number fields. (English) Zbl 0823.11064

Let \(K/k\) be a \(\mathbb{Z}_ p\)-extension of a number field \(k\). Denote by \(A_ n\) the \(p\)-part of the ideal class group of \(k_ n\), the \(n\)-th layer of \(K/k\) \((n\geq 0)\). For \(m\geq n\), let \(H_{n,m}\) denote the kernel of the map \(A_ n\to A_ m\) induced by the natural inclusion \(k_ n\to k_ m\). Assume that only one prime of \(k\) lies above \(p\) and that this prime is totally ramified in \(K/k\). The author’s main result states that (i) if \(H_{0,n}= A_ 0\) for some \(n\geq 1\), then \(| A_ m |= | A_ n|\) for all \(m\geq n\); (ii) if \(| A_{n+1} |= | A_ n|\) for some \(n\geq 0\) and the exponent of \(A_ n\) is \(p^ r\), then \(H_{n, n+r}= A_ n\). (Here \(| A_ n|\) denotes the cardinality of \(A_ n\).) This continues the study of the relationship between \(| A_ n |\) and \(H_{n,m}\), initiated by R. Greenberg [Am. J. Math. 98, 263-284 (1976; Zbl 0334.12013)]. The author also obtains a lower bound for \(| A_ n|\) and discusses, without any restriction on the decomposition of \(p\), how the behaviour of \(| A_ n|\) and \(\text{rank} (A_ n)\) are linked to the vanishing of the invariants \(\mu\) and \(\lambda\) of \(K/k\).

MSC:

11R23 Iwasawa theory
11R29 Class numbers, class groups, discriminants

Citations:

Zbl 0334.12013
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References:

[1] R. Greenberg: On the Iwasawa invariants of totally real number fields. Amer. J. Math., 98, 263-284 (1976). JSTOR: · Zbl 0334.12013
[2] K. Iwasawa: On ^-extensions of algebraic number fields. Ann. of Math., 98, 246-326 (1973). JSTOR: · Zbl 0285.12008
[3] S. Maki: The determination of units in real cyclic sextic fields. Lect, Notes in Math., vol. 797, Springer-Verlag, Berlin, Heidelberg, New York (1980). · Zbl 0423.12006
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