A remark on the capitulation problem (Q1677559)

Let \(\ell \neq \ell'\) be prime numbers with \(\ell \equiv -1 \mod (4)\) and \(\ell' \equiv 3 \mod (8)\), and put \(k = \mathbb Q(\sqrt{\ell\ell'})\). For \(n \in \mathbb N\), let \(k_n\) denote the \(n\)-th layer of the cyclotomic \(\mathbb Z_2\)-extension of \(k\), and \(M_n \neq k_n\) denote that quadratic subfield of \(k_{n+1}\), which does not belong to the cyclotomic \(\mathbb Z_2\)-extension of \(\mathbb Q\). The author determines the \(2\)-rank of the ideal class groups of \(M_n\) and \(k_{n+1}\), shows that both fields have the same Hilbert \(2\)-class field and an infinite Hilbert \(2\)-class tower. Furthermore, he determines the number of \(2\)-ideals classes of \(M_n\) which capitulate in \(k_{n+1}\), which depends on the \(2\)-valuation of \(\ell+1\). In case of \(\ell\) being a Mersenne prime, this gives an impressive large number of ideal classes, which capitulate in the quadratic extension \(k_{n+1}/M_n\).