Capitulation in the cyclotomic \(\mathbb{Z}_2\) extension of CM number fields. (English) Zbl 1462.11099

Summary: Let \(\mathbb{K}_n\) be the intermediate steps in the cyclotomic \(\mathbb{Z}_p\)-extension of a CM number field \(\mathbb{K}\). For \(p\neq 2\) the minus part of the \(p\)-class groups is given by \(A_n^-=\frac{1}{2}(1-j)A_n\). We will give a new definition of the minus part for \(p=2\) and prove that there is no finite submodule in \(\lim_{\infty\leftarrow n}A_n^-\). Furthermore we show that \(\mu=0\) if and only if \(\mu^-=0\) in this new definition.


11R23 Iwasawa theory
11R18 Cyclotomic extensions
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[1] Washington, L. C., Introduction to Cyclotomic Fields, (1982), Springer · Zbl 0484.12001
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