On the structure of the Galois group of the maximal pro-\(p\) extension with restricted ramification over the cyclotomic \(\mathbb{Z}_p\)-extension (Q784836)

Let \(p\) be a prime and let \(k_{\infty}\) be the cyclotomic \(\mathbb Z_p\)-extension of a number field \(k\). Let \(S\) be a finite set of primes which does not contain \(p\). The author investigates the structure of \(G_S(k_{\infty})\), the Galois group over \(k_{\infty}\) of the maximal pro-\(p\)-extension of \(k_{\infty}\) which is unramified outside \(S\). In particular, he studies the question whether \(G_S(k_{\infty})\) is a non-abelian free pro-\(p\)-group. Set \(G(k_{\infty}) := G_{\emptyset}(k_{\infty})\) and let \(X(k_{\infty})\) be its abelianization. It is expected that \(G(k_{\infty})\) is never a non-abelian free pro-\(p\)-group. Some still conjectural evidence for this is the following result of \textit{S. Fujii} [Acta Arith. 149, No. 2, 101--110 (2011; Zbl 1250.11094)]: Suppose that \(p\) splits completely in \(k\). Then \textit{R. Greenberg}'s [Adv. Stud. Pure Math. 30, 335--385 (2001; Zbl 0998.11054)] generalized conjecture fails for \(k\) at \(p\) if \(G(k_{\infty})\) is a non-abelian free pro-\(p\)-group. In fact, Fujii has shown that Greenberg's conjecture implies that \(G(k_{\infty})\) is either isomorphic to \(\mathbb Z_p\) or has derived depth at most \(1\). \par In the first part of the paper under review the author gives sufficient conditions for the non-freeness of \(G(k_{\infty})\) for imaginary quadratic \(k\) and odd \(p\), mainly in terms of \(\lambda\)-invariants of \(X(k_{\infty})\) and \(X(F_{\infty})\), where \(F\) is an extension of \(\mathbb Q\) of degree \(p\) that is contained in an unramified cyclic extension \(K\) of \(k\). The existence of \(K\) is guaranteed by one of the conditions on \(k\), namely that the class number of \(k\) is divisible by \(p\). Moreover, \(p\) is assumed to be non-split in \(k\). In the second part of the paper, \(p\) is an arbitrary prime and the number field \(k\) is assumed to be totally real. The main result is the following variant of Fujii's theorem. If \(G_S(k_{\infty})\) is a non-abelian free pro-\(p\)-group, then there is a totally real number field (in general different from \(k\)) such that Greenberg's conjecture fails. More generally, there exists a counterexample for Greenberg's conjecture if there is an infinite Galois extension of \(k_{\infty}\) that is unramified outside \(S\) and whose Galois group is a free pro-\(p\)-group of generator rank at least \(2\) or a Demuškin group of generator rank at least \(3\). The paper also contains some computational evidence for the non-freeness of \(G(k_{\infty})\) for imaginary quadratic \(k\) and \(p=3\).