On a quotient of the unramified Iwasawa module over an abelian number field. II.
Let \(p\) be an odd prime number, \(k\) an abelian extension of \(\mathbb Q\) containing a primitive \(p\)-th root of unity \(\zeta_p\), and let \(k_\infty/k\) be the cyclotomic \(\mathbb Z_p\)-extension of \(k\). Let \(L\) be the maximal unramified pro-\(p\) abelian extension of \(k_\infty\), and \(L'\) the maximal subextension of \(L/k_\infty\) in which all primes above \(p\) split completely. Finally, let \(N/k_\infty\) and \(N'/k_\infty\) denote the pro-\(p\) abelian extensions generated by all \(p^n\)-th roots of units (or \(p\)-units) in \(k_\infty\). Put \({\mathcal Y} = \text{Gal}(N \cap L/k_\infty)\) and \({\mathcal Y}' = \text{Gal}(N' \cap L'/k_\infty)\), and let \({\mathcal Y}^-\) and \({{\mathcal Y}'}^-\) denote the odd parts of these \(\mathbb Z_p[\text{Gal}(k/\mathbb Q)]\)-modules. In part I [J. Number Theory 88, No. 1, 175--190 (2001; Zbl 0972.11104)] the author proved that the \(\mathbb Z_p\)-torsion subgroup of \({\mathcal Y}^-\) is isomorphic to a subquotient of the plus class group of \(k_\infty\) as a \(\mathbb Z_p[\text{Gal}(k/\mathbb Q)]\)-module. Here, the same is proved for the \(\mathbb Z_p\)-torsion subgroup of \({{\mathcal Y}'}^-\).
scientific article; zbMATH DE number 1868177