\(K_2\) and the Greenberg conjecture in multiple \(\mathbb Z_p\)-extensions (Q819869)

The ``generalized Greenberg conjecture'' (GG) affirms that the usual Iwasawa module \(X(\widetilde K)\) attached to the compositum \(\widetilde K\) of all \(\mathbb Z_p\)-extensions of a given number field \(K\) is pseudo-null over the Iwasawa algebra \(\widetilde \Lambda\) (which is defined as \(\mathbb Z_p[[G(\widetilde K/K)]]\)). Pseudo-nullity means that the module has support of codimension at least 2. The Galois group \(G(\widetilde K/K)\) is isomorphic to \(\mathbb Z_p^d\) for some \(d\). If \(K\) is totally real and Leopoldt's conjecture holds for \(K\) and \(p\), then \(d=1\); we get back Greenberg's usual conjecture. (A finitely generated \(\mathbb Z_p[[T]]\)-module is pseudo-null iff it is finite.) \textit{W. G. McCallum} [Am. Math. J. 123, No. 5, 909--939 (2001; Zbl 1020.11068)] proposed a criterion for (GG) to hold, which would include for instance the field \(K\) mentioned at the end of this review. As explained in the paper under review, his argument seems to contain a gap which has not be fixed so far. (Note that the reference number in line 7 of p .670 should be [15], not [13]). The authors therefore change the strategy somewhat and come up with a slightly different approach. They proceed in three steps, the first two of which are essentially known from the work of McCallum: (1) (GG) is equivalent to the triviality of the \(\widetilde \Lambda\)-torsion submodule \(T=\text{tors}_{\widetilde\Lambda}{\mathcal X}(\widetilde K)\), where \({\mathcal X}(M)\) denotes the Galois group of the maximal abelian pro-\(p\)-extension of \(M\) unramified outside \(p\). (2) One looks at the codescent map \(\delta\) from \(T_{\widetilde \Gamma}\) to \({\mathcal X}(K)\) and proves its injectivity. (3) One proves that the map \(\delta\) is null. This together with (2) proves the vanishing of \(T\) (even before taking coinvariants), and (GG) follows by virtue of (1). Actually one may replace \(\mathcal X\) and \(\delta\) by twisted versions \({\mathcal X}^{(i)}\) and \(\delta^{(i)}\); the main result uses \(i=-2\), and this leads to the connection with capitulation problems for \(K_2\), for the reason that \(\delta^{(i)}\) is shown (in Theorem 3.1) to be dual to a ``capitulation map'' for certain ``twisted class groups''. The final result then says that four hypotheses (a)--(d) taken together imply (GG) for \(K\). Here they are: (a) \(K\) contains the \(p\)-th roots of unity. (b) The \(\mathbb Z_p\)-torsion submodule of \({\mathcal X}(K)\) is cyclic. (c) The Leopoldt conjecture holds for \(K\) and \(p\). (d) The \(p\)-part of the tame kernel \(K_2(O_K')\{p\}\) capitulates in \(\widetilde K\). -- The proof is intricate, using a fair amount of cohomological calculation and some general algebraic lemmas concerning ranks of \(\widetilde\Lambda\)-modules. With the aid of a calculation of McCallum and Sharifi, the authors are then able to show that all conditions are satisfied for \(p=37\) and \(K=\mathbb Q(\zeta_p)\). (Regular primes \(p\) are of no interest in this context.) This is interesting progress on a difficult subject. The paper concludes with a useful appendix discussing slightly different formulations of the generalised Greenberg conjecture.