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The class group of \(\mathbb{Z}_p\)-extensions over totally real number fields. (English) Zbl 0896.11044

Let \(k\) be a totally real number field, \(p\) an odd prime, and \(k_\infty/k\) a \(\mathbb{Z}_p\)-extension. Let \(L_\infty/k_\infty\) be the maximal unramified abelian pro-\(p\)-extension, and \(M_\infty/k_\infty\) the maximal abelian pro-\(p\)-extension unramified outside \(p\). The first theorem of this paper states that if \(p\) splits completely in \(k\) and is totally ramified in \(k_\infty/k\), then the following conditions are equivalent: i) \(L_\infty \neq k_\infty\); ii) the \(p\)-class number of the \(n\)th layer \(k_n\) of \(k_\infty/k\) is nontrivial for every \(n \geq 1\); iii) \(\zeta_p(0)\) is a \(p\)-adic integer, where \(\zeta_p(s)\) is the \(p\)-adic zeta function of \(k\). Now assume in addition that Leopoldt’s conjecture holds for \(k\) and \(p\), then the author’s second theorem shows the equivalence of the following three assertions: i) there is nontrivial capitulation in \(k_\infty/k_1\); ii) there is nontrivial capitulation in \(k_\infty/k_n\) for some \(n \geq 0\); iii) \(M_\infty \neq L_\infty\). If, in addition, the \(p\)-class groups of the \(k_n\) are cyclic and if the \(\mathbb{Z}_p\)-rank of \(M_\infty/k_\infty\) is at least \(2\), then Greenberg’s conjecture for \(k_\infty/k\) is true.

MSC:

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