A consequence of Greenberg's generalized conjecture on Iwasawa invariants of \(\mathbb{Z}_p\)-extensions (Q344130)

Let \(p\) be a prime and let \(K\) be an algebraic extension of \(\mathbb Q\). Let \(X(K)\) be the Galois group over \(K\) of the maximal unramified abelian pro-\(p\)-extension of \(K\). If \(K\) is a \(\mathbb Z_p^d\)-extension of some number field \(k\), then \(X(K)\) is a finitely generated torsion \(\Lambda(K/k)\)-module, where \(\Lambda(K/k)\) denotes the Iwasawa algebra \(\mathbb Z_p[[\mathrm{Gal}(K/k)]]\).NEWLINENEWLINENow let \(\tilde k\) be the compositum of all \(\mathbb Z_p\)-extensions of \(k\). Then Greenberg's generalized conjecture asserts that \(X(\tilde k)\) is pseudo-null as a \(\Lambda(\tilde k/k)\)-module (see \textit{R. Greenberg} [Adv. Stud. Pure Math. 30, 335--385 (2001; Zbl 0998.11054)]). We will henceforth assume that this conjecture holds.NEWLINENEWLINEIf \(k\) is an imaginary quadratic field, \textit{M. Ozaki} [Adv. Stud. Pure Math. 30, 387--399 (2001; Zbl 1002.11078)] has shown that then the following holds for all but finitely many \(\mathbb Z_p\)-extensions \(K\) over \(k\): If one prime of \(k\) above \(p\) does not split in \(K/k\), then \(\mu(K/k) = 0\) and \(\lambda(K/k) = s\) (where \(s=1\) if \(p\) splits in \(k\), and \(s=0\) otherwise). Here, \(\mu(K/k)\) and \(\lambda(K/k)\) are the usual \(\lambda\)- and \(\mu\)-invariants of the Iwasawa module \(X(K)\).NEWLINENEWLINEIn the article under review, the author (partially) generalizes Ozaki's result to arbitrary number fields \(k\).NEWLINENEWLINERecall that the set \(\mathcal E(k)\) of all \(\mathbb Z_p\)-extensions of \(k\) is equipped with a compact Hausdorff topology. Two \(\mathbb Z_p\)-extensions \(K\) and \(K'\) are close if \([K \cap K' : k] = p^n\) with \(n\) large. For \(K \in \mathcal E(k)\) let \(s(K/k)\) be the \(\mathbb Z_p\)-rank of the \(\mathrm{Gal}(K/k)\)-coinvariants of \(X(K)\). Moreover, let \(s(k)\) be the minimum of the \(s(K/k)\), where \(K\) runs through all \(\mathbb Z_p\)-extensions of \(k\) such that every \(p\)-adic place of \(k\) ramifies in \(K/k\). For instance, one can show that \(s(k) = 0\) if \(p\) does not split in \(k\). If \(p\) splits completely in \(k\), one has \(s(k) = d(k) - 1\), where \(d(k)\) is the \(\mathbb Z_p\)-rank of \(\mathrm{Gal}(\tilde k/k)\). In particular, when \(p\) splits completely in an imaginary abelian field \(k\), then \(s(k) = [k:\mathbb Q] / 2\).NEWLINENEWLINEIn a similar way the author defines a second integer \(s'(k)\) which conjecturally always vanishes. Assuming \(s'(k) = 0\) and Greenberg's generalized conjecture, the author shows the following for generic \(\mathbb Z_p\)-extensions of \(k\) (where `generic' means that there is a closed subset \(E\) of \(\mathcal E(k)\) of measure \(0\) which contains the set of all \(K\) for which the following statement is not true):NEWLINENEWLINEIf every \(p\)-adic place of \(k\) does not split in \(K/k\), then \(\mu(K/k) = 0\) and \(\lambda(K/k) = s(k)\).