×

Algorithmic approach to logarithmic class groups. (Approche algorithmique du groupe des classes logarithmiques.) (French) Zbl 0930.11079

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 [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.

MSC:

11R29 Class numbers, class groups, discriminants
11Y40 Algebraic number theory computations
11R70 \(K\)-theory of global fields

Software:

PARI/GP
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. Browkin, H. Gangl, Table of tame and wild kernels of quadratic imaginary number fields of discriminants >−5000 (conjectural values), Institut für Experimentelle Mathematik, 20; J. Browkin, H. Gangl, Table of tame and wild kernels of quadratic imaginary number fields of discriminants >−5000 (conjectural values), Institut für Experimentelle Mathematik, 20 · Zbl 0919.11079
[2] Browkin, J.; Schinzel, A., On Sylow 2-subgroups of \(K_2_F)\) for quadratic number fields \(F\), J. Reine Angew. Math., 331, 104-113 (1982) · Zbl 0493.12013
[3] Cohen, H., A Course in Computational Algebraic Number Theory. A Course in Computational Algebraic Number Theory, Graduate Text in Mathematics, 138 (1993), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York · Zbl 0786.11071
[4] Cohen, H.; Diaz y. Diaz, F.; Olivier, M., Algorithmic methods for finitely generated abelian groups, Symbolic Comput. (1997)
[5] Candiotti, A.; Kramer, K., On the 2-Sylow subgroup of the Hilbert kernel of \(K_2\), Acta Arith., L II, 49-65 (1989) · Zbl 0705.19005
[6] Diaz y. Diaz, F.; Olivier, M., Algorithmique Algébrique dans les Corps de Nombres, État de la Recherche en Algorithmique Arithmétique (1995)
[7] Hafner, J.; McKurley, K., A rigorous sous exponential algorithm for computation of class groups, J. Ann. Math. Soc., 2, 837-850 (1989) · Zbl 0702.11088
[8] Jaulent, J.-F., Noyau universel et valeurs absolues, Journées Arithmétiques de Marseille-Luminy, Astérisque, 198-199-200, 187-207 (1991)
[9] Jaulent, J.-F., Sur le noyau sauvage des corps de nombres, Acta Arith., 67, 335-348 (1994) · Zbl 0835.11042
[10] Jaulent, J.-F., Classes logarithmiques des corps de nombres, J. Théor. Nombres Bordeaux, 6, 307-327 (1994)
[11] J.-F. Jaulent, Classes logarithmiques des corps de nombres totalement réels; J.-F. Jaulent, Classes logarithmiques des corps de nombres totalement réels
[12] Bernardi, D.; Batut, C.; Cohen, H.; Olivier, M., User’s Guide to PARI-GP, Version 1.39. User’s Guide to PARI-GP, Version 1.39, Publ. Université Bordeaux I (1991)
[13] Soriano, F., Classes logarithmiques ambiges des corps quadratiques, Acta Arith., 78, 201-219 (1997) · Zbl 0869.11081
[14] Soriano, F., Sur les classes logarithmiques des (CM)-extensions, C. R. Acad. Sci. Paris Sér. I Math., 324, 737-739 (1997) · Zbl 0887.11045
[15] Thomas, H., Trivialité du 2-rang du noyau hilbertien, J. Théor. Nombres Bordeaux, 6, 459-483 (1994) · Zbl 0822.11079
[16] Zran Zankoe, T. S., Noyau des valeurs absolues 3-adiques (1991)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.