On a bound of \(\lambda\) and the vanishing of \(\mu\) of \(\mathbb{Z}_p\)-extensions of an imaginary quadratic field
Let \(k\) be an imaginary quadratic field and \(\tilde{k}\) the compositum of all the \(\mathbb{Z}_p\)-extensions of \(k\) (\(p\) an odd prime): it is well known that \(\text{Gal}(\tilde{k}/k)\simeq \mathbb{Z}_p^2\) so there are infinitely many different \(\mathbb{Z}_p\)-extensions of \(k\). For any \(\mathbb{Z}_p\)-extension \(k_\infty /k\), let \(\lambda(k_\infty)\) and \(\mu(k_\infty)\) be the associated Iwasawa \(\lambda\) and \(\mu\)-invariants. The paper deals with their behaviour as \(k_\infty\) varies inside \(\tilde{k}\). Whenever \(\mu(k_\infty)=0\) one has that the Iwasawa module \(X(k_\infty):= \text{Gal}(L(k_\infty)/k_\infty)\) is a free \(\mathbb{Z}_p\)-module of rank \(\lambda(k_\infty)\) (where \(L(k_\infty)\) is the maximal abelian unramified pro-\(p\)-extension of \(k_\infty\,\)). Thus the author provides conditions (usually on the behaviour of the prime \(p\) in the extension \(k/\mathbb{Q}\) and on \(\lambda(k_{cyc})\), the \(\lambda\)-invariant of the cyclotomic \(\mathbb{Z}_p\)-extension of \(k\)) which ensure (or enable him to prove) the vanishing of \(\mu(k_\infty)\) and then studies \(X(k_\infty)\) as a module over the Iwasawa algebra \(\Lambda:=\mathbb{Z}_p[[\text{Gal}(\tilde{k}/k)]]\simeq \mathbb{Z}_pS,T\), using the following exact sequence (of \(\mathbb{Z}_pGal(k_\infty/k)\)-modules) \[ 0\rightarrow Y(k_\infty) \rightarrow X(k_\infty) \rightarrow \text{Gal}(L(k_\infty) \cap \tilde{k}/k_\infty) \rightarrow 0 \] (taken from \textit{M. Ozaki} [Adv. Stud. Pure Math. 30, 387--399 (2001; Zbl 1002.11078)], where \(Y(k_\infty)\) represents the \(\text{Gal}(\tilde{k}/k_\infty)\)-coinvariants of \(X(\tilde{k})\,\)). Let \(S\) be the variable corresponding to a topological generator of \(\text{Gal}(\tilde{k}/k_{\text{cyc}})\), then, using the structure theorem for finitely generated torsion \(\Lambda\)-modules, one finds a series \(f(S)\in \mathbb{Z}_pS\) and a sequence \[ \Lambda/((1+S)^\alpha(1+T)^\beta-1,T-f(S)) \rightarrow X(k_\infty) \rightarrow \text{Gal}(L(k_\infty) \cap \tilde{k}/k_\infty) \rightarrow 0 \;, \] where \((1+S)^\alpha(1+T)^\beta-1\) corresponds to a topological generator of \(\text{Gal}(\tilde{k}/k_\infty)\) (with \((\alpha,\beta)\in \mathbb{Z}_p^2-p\mathbb{Z}_p^2\,\)). Assuming that \(p\) splits in \(k\) and \(\lambda(k_{\text{cyc}})=2\) (resp. \(p\) does not split in \(k\) and \(\lambda(k_{\text{cyc}})=1\)), with a careful study of the series \(f(S)\) and of the ideals \(((1+S)^\alpha(1+T)^\beta-1,T-f(S),p)\), the author provides the upper bound \(\lambda(k_\infty)\leq 2\) (resp. \(\lambda(k_\infty)\leq 1\)) for all \(\mathbb{Z}_p\)-extensions disjoint from the anticyclotomic one and in which both primes over \(p\) ramify. When \(p\) splits in \(k\) the larger bound \(\lambda(k_\infty)\leq p\) is given for all \(\mathbb{Z}_p\)-extensions but with the additional hypotheses of having \(L(k)\subseteq \tilde{k}\) and of having the decomposition group of \(p\) of index \(p\) in \(\text{Gal}(\tilde{k}/k)\).
scientific article; zbMATH DE number 6152434