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On the mu and lambda invariants of the logarithmic class group. (English) Zbl 1475.11201

Let \(K\) be a number field and \(l\) a prime number. Let \(K^c\) be the \({\mathbb Z}_l\)-cyclotomic extension of \(K\) and let \(K^{lc}\) be the maximal abelian pro-\(l\)-extension of \(K\) which splits completely over \(K^c\). The logarithmic class group \(\widetilde{Cl}_K\) of \(K\) is the \({\mathbb Z}_l\)-module isomorphic to \(\mathrm{Gal}(K^{lc}/K^c)\). J.-F. Jaulent proved that the finiteness of \(\widetilde{Cl}_K\) is equivalent to the Gross-Kuz’min conjecture [Ann. Math. Qué. 41, No. 1, 119–140 (2017; Zbl 1432.11161)].
The aim of this paper is the study of logarithmic class groups in the spirit of Iwasawa’s work for class groups. The main result of this paper is the analogue of the results of Iwasawa for class groups of \({\mathbb Z}_l\)-extensions. Namely, let \(K_{\infty}/K\) be a \({\mathbb Z}_l\) extension and assume that the Gross-Kuz’min conjecture is valid along the \({\mathbb Z}_l\)-extension \(K_{\infty}\). Let \(K_n\) be the \(n\)-th layer of \(K_{\infty}/K\), \(\widetilde{Cl}_n\) the logarithmic class group of \(K_n\) and let \(l^{\tilde{e}_n}\) be its order. Then, there exist integers \(\tilde \lambda, \tilde \mu \geq 0\) and \(\tilde \nu\) such that \(\tilde{e}_n=\tilde \mu l^n+\tilde \lambda n+\tilde \nu\) for \(n\) big enough.
This result was proved by J.-F. Jaulent in [Publ. Math. Fac. Sci. Besançon, Théor. Nombres Années 1984/85-1985/86, No. 1, 349 pp. (1986; Zbl 0601.12002)] when \(K_{\infty}/K\) is the cyclotomic \({\mathbb Z}_l\)-extension. The non-cyclotomic case is the content of Theorem 4.3. Additionally, in Section 5, the athor provides numerical examples, in both the cyclotomic and non-cyclotomic cases, of logarithmic class groups in the first layers of \({\mathbb Z}_l\)-extensions and also explicitly compute the \(\tilde\mu, \tilde\lambda, \tilde\gamma\) logarithmic invariants.

MSC:

11R23 Iwasawa theory
11R37 Class field theory

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References:

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