Algorithmic approach to logarithmic class groups
type: Item
instance of: scholarly article
title: Algorithmic approach to logarithmic class groups
zbMATH Open document ID: 0930.11079
author: K. Appert
published in: Journal of Number Theory
publication date: 2000-02-14T00:00:00Z
review text: The authors describe an efficient algorithm for computing the logarithmic class group of an algebraic number field \(K\) in connection with the wild kernel of \(K_2(k)\). Logarithmic classes were introduced by the reviewer [\textit{J. F. Jaulent}, J. Théor. Nombres Bordx. 6, 307-327 (1994; Zbl 0827.11064)] and so-called because their definition involves \(\ell\)-adic valuations using the Iwasawa logarithm of local norms. By \(\ell\)-adic class field theory the logarithmic \(\ell\)-class group \(\widetilde{Cl}_k\) correspond to the Galois group \(\text{Gal}(K^{lc}/K^c)\) over the cyclotomic \(\mathbb{Z}_\ell\)-extension \(K^c\) of the maximal abelian pro-\(\ell\)-extension \(K^k\) of \(K\) which is locally trivial over \(K^c\). But its main interest is the canonical isomorphism \[ \mu_{\ell}\otimes\widetilde{Cl}_K\simeq H_2(K)/H_2(K)^\ell \] given by the reviewer [Acta Arith. 67, 335-348 (1994; Zbl 0835.11042)] where \(H_2(K)\) is the Hilbert kernel in \(K_2(K)\) and \(K\) is assumed to contain the \(2\ell\)-th roots of unity. In this paper the authors describe from an algorithmic point of view the structure of the logarithmic \(\ell\)-class group \(\overline{Cl}_K\) for finite Galois extensions \(K\) of \(\mathbb{Q}\), and they illustrate their method using the PARI package by performing the computation of \(\widetilde{Cl}_K\) for biquadratic number fields \(\mathbb{Q}[\sqrt{-1},\sqrt d]\) in the case \(\ell=2\) and \(\mathbb{Q}[\sqrt{-3},\sqrt d]\) in the case \(\ell=3\) for \(d\leq 2000\). As explained above, their results give also the \(\ell\)-rank of the Hilbert kernel for these fields.
reviewed by: Jean-François Jaulent
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11Y40
Mathematics Subject Classification ID: 11R70
zbMATH DE Number: 1310080
zbMATH Keywords: \(\ell\)-rank of the Hilbert kernel
zbMATH Keywords: efficient algorithm
zbMATH Keywords: logarithmic class group
zbMATH Keywords: PARI package
zbMATH Keywords: computation
zbMATH Keywords: biquadratic number fields
describes a project that uses: PARI/GP
MaRDI profile type: Publication
full work available at URL: jnth.1998.2344
OpenAlex ID: W2010585143
cites work: Tame and wild kernels of quadratic imaginary number fields
cites work: On Sylow 2-subgroups of K2OF for quadratic number fields F.
cites work: Q3139838
cites work: On the 2-Sylow subgroup of the Hilbert kernel of $K_{2}$ of number fields
cites work: A Rigorous Subexponential Algorithm For Computation of Class Groups
cites work: Sur le noyau sauvage des corps de nombres
cites work: Classes logarithmiques ambiges des corps quadratiques
cites work: Sur les classes logarithmiques des CM -extensions
cites work: Triviality of the 2-rank of the Hilbert kernel
DOI: 10.1006/JNTH.1998.2344
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description: scientific article; zbMATH DE number 1310080