Some problems on \(p\)-class field towers (Q2454944)

Let \(p\) be an odd prime number, and let \(K\) denote a CM-field, i.e., a totally complex quadratic extension \(K/K^+\) of a totally real algebraic number field \(K^+\). Let \(K_\infty\) denote the cyclotomic \(\mathbb Z_p\)-extension of \(K\), and \(K_n\) its \(n\)-th layer. Let us denote the \(p\)-class group of a field \(K\) by \(A_K\), and let \(\widetilde{G}_\infty\) denote the Galois group of the maximal unramified pro-\(p\) extension \(\widetilde{L}/K_\infty\). The main result of this article is the following: assume that \(\widetilde{G}_\infty\) is a free pro-\(p\) group with rank \(\lambda \geq 2\), that \(p\) does not split in \(K_\infty/\mathbb Q\), and that the class number of \(K^+\) is not divisible by \(p\). Then \(\widetilde{L} K_1/K_1\) is an infinite extension if \(p \geq 5\) and \(A_K\) has rank \(2\), or if \(p = 3\) and \(A_K\) is isomorphic to \(\mathbb Z/3^a\mathbb Z \oplus \mathbb Z/3^b\mathbb Z\) for integers \(a, b \geq 2\). For fields where the \(p\)-class group \(A_K\) has rank \(\geq 3\), this result is classical [\textit{H. Kisilevsky} and \textit{J. Labute}, `` On a sufficient condition for the p-class tower of a CM-field to be infinite'', Théorie des nombres, C. R. Conf. Int., Québec/Can. 1987, 556--560 (1989; Zbl 0696.12009)]. The authors also determine \(\widetilde{G}_\infty\) for a class of fields satisfying several rather strong conditions.