The second syzygy of the trivial \(G\)-module, and an equivariant main conjecture
In this paper, the authors determine the (initial) Fitting ideal of the \(S\)-ramified Iwasawa module attached to an abelian extension \(K/k\) of totally real number fields. This builds upon previous work of the first two authors [St. Petersbg. Math. J. 27, No. 6, 941--965 (2016; Zbl 1359.11087); Algebra Anal. 27, No. 6, 117--149 (2015; Zbl 1359.11087); Tokyo J. Math. 39, No. 3, 619--642 (2017; Zbl 1416.11158)] on the same topic. Let \(K_{\infty}\) be the cyclotomic \(\mathbb{Z}_p\)-extension of \(K\) and set \(\mathcal{G} := \mathrm{Gal}(K_{\infty}/k)\). We can assume without loss of generality that \(\mathcal{G} = G \times \Gamma\), where \(G = \mathrm{Gal}(K/k)\) and \(\Gamma \simeq \mathbb{Z}_p\). Let \(S\) be a finite set of places of \(k\) containing all archimedean places and all places that ramify in \(K_{\infty}/k\). The Iwasawa module in question is \(X_S\), the Galois group over \(K_{\infty}\) of the maximal abelian pro-\(p\)-extension of \(K_{\infty}\) which is unramified outside \(S\). This is a finitely generated \(\Lambda(\mathcal{G}) := \mathbb{Z}_p[[\mathcal{G}]]\)-module which is torsion as \(\Lambda(\Gamma)\)-module. In the work mentioned above it was shown that \[ \mathrm{Fitt}_{\Lambda(\mathcal{G})}(X_S) = \mathfrak{A} \Theta_S, \] where \(\Theta_S\) is an equivariant \(p\)-adic \(L\)-function and \(\mathfrak{A}\) is a certain \(\Lambda(\mathcal{G})\)-ideal. It was assumed throughout that \(X_S\) is finitely generated as \(\mathbb{Z}_p\)-module (that is the conjectural vanishing of its \(\mu\)-invariant). However, this was only required to ensure the validity of the equivariant Iwasawa main conjecture, which is meanwhile known to hold unconditionally in this situation by work of \textit{H. Johnston} and the reviewer [``An unconditional proof of the abelian equivariant Iwasawa main conjecture and applications'', Preprint, \url{arXiv:2010.03186}]. The ideal \(\mathfrak{A}\) is purely algebraic in nature, i.e. it only depends upon \(G\) and not on the underlying extension \(K/k\). The description of \(\mathfrak{A}\) is rather involved and it remained open to determine it completely. This is carried out in the paper under review. In particular, one has to compute the higher Fitting ideals of a second syzygy of \(\mathbb{Z}\) (or \(\mathbb{Z}_p\) considered as a \(G\)-module with trivial action) and hence minors of a huge relation matrix. For the entire collection see [Zbl 1462.11006].
scientific article; zbMATH DE number 7373075