On the isomorphism classes of Iwasawa modules with \(\lambda = 3\) and \(\mu = 0\)
Let \(E\) be a finite extension of \(\mathbb Q_p\) with ring of integers \({\mathcal O}_E\) and uniformizer \(\pi\). Let \(\Lambda_E={\mathcal O}_ET\). Let \(f(T)=(T-\alpha)(T-\beta)(T-\gamma)\), where \(\alpha, \beta, \gamma\) are distinct elements of \(\pi{\mathcal O}_E\). The author classifies all finitely generated \(\Lambda_E\)-torsion modules that are free over \({\mathcal O}_E\) and have characteristic polynomial \(f(T)\). Such a module \(M\) injects into \({\mathcal O}_E\times{\mathcal O}_E\times{\mathcal O}_E\) via the map \(M\hookrightarrow \Lambda_E/(T-\alpha)\oplus \Lambda_E/(T-\beta)\oplus \Lambda_E/(T-\gamma)\). The action of \(T\) on \({\mathcal O}_E^3\) is multiplication by \(\alpha, \beta, \gamma\) on the respective components. The author shows that \(M\) is \(\Lambda_E\)-isomorphic to a module \(M(m,n,x)\) generated over \({\mathcal O}_E\) by \((1,1,1), (0,\pi^m,x), (0,0,\pi^n)\), with \(m\) and \(n\) satisfying the conditions \(0\leq m\leq \text{ord}_E(\beta-\alpha)\), \(0\leq n\leq \text{ord}_E(\gamma-\beta)+\text{ord}_E(x)\), and \(n\leq \text{ord}_E((\gamma-\alpha)-(\beta-\alpha)\pi^{-m}x)\). Moreover, \(M(m,n,x)\) is \(\Lambda_E\)-isomorphic to \(M(m',n',x')\) if and only if \(m=m', n=n'\), and \(x\equiv x' \pmod{\pi^n{\mathcal O}_E}\). Similar results have been obtained independently by \textit{C.\ Franks} [Classifying \(\Lambda\)-modules up to isomorphism and applications to Iwasawa theory. Phoenix: Arizona State University (PhD Thesis) (2011)]. The present result extends work of \textit{M. Koike} [J. Math. Soc. Univ. Tokyo 6, No. 2, 371--396 (1999; Zbl 1052.11072)] and \textit{H. Sumida} [J. Math. Soc. Japan 49, No. 4, 689--711 (1997; Zbl 0907.11038)], who treated the case where \(f(T)\) has degree \(2\). Sumida [Abh. Math. Semin. Univ. Hamb. 70, 113--117 (2000; Zbl 1016.11049)] also treated certain cases where \(f(T)\) has degree 3. The author uses the results of the present paper to determine the \(\Lambda\)-isomorphism classes of Iwasawa modules arising from the ideal class groups for \(\mathbb Z_3\)-extensions of imaginary quadratic fields \(\mathbb Q(\sqrt{-d})\) with \(1<d<10^5\) and \(d\not\equiv 2\pmod 3\) and \(\lambda_3=3\).
scientific article; zbMATH DE number 6377176
scientific article