Greenberg's conjecture and units in multiple \(\mathbb Z_p\)-extensions (Q2762639)

Let \(K\) be a CM number field, \(p\) an odd prime number, \(\widetilde K\) the composition of all \(\mathbb Z_p\)-extensions of \(K\), \(\wedge= \mathbb Z_p [[G(\widetilde K/K)]]\). Greenberg's generalized conjecture (GGC) states that the Galois group over \(\widetilde K\) of the maximal Abelian unramified pro-\(p\)-extension of \(\widetilde K\) is pseudo-null as a \(\wedge\)-module. If \(K\) is totally real and satisfies Leopoldt's conjecture, then \(\widetilde K\) is just the cyclotomic \(\mathbb Z_p\)-extension of \(K\) and one recovers Greenberg's usual conjecture (GC). In this paper, modulo some (perhaps unnecessary) restrictions on ramification, (GGC) is shown to be equivalent to the vanishing of the \(\wedge\)-torsion submodule of the Galois group over \(\widetilde K\) of the maximal Abelian \(p\)-ramified pro-\(p\)-extension of \(\widetilde K\) [see also \textit{A. Lannuzel} and \textit{T. Nguyen Quang Do}, Manuscr. Math. 102, 187--209 (2000; Zbl 0983.11063)]. This version of (GGC) is then proved for \(K=\mathbb Q (\xi_p)\), assuming that the \(p\)-class group of \(K\) is of order \(p\) (which implies Vandiver's conjecture), as well as additional very restrictive conditions on ramification.