Signed logarithmic classes of number fields. (Classes logarithmiques signées des corps de nombres.) (French) Zbl 0997.11095

The author defines a signed logarithmic 2-class group of a number field as the actual analogue in the logarithmic context of the usual restricted ideal 2-class group.
Logarithmic classes were introduced by the author [J. Théor. Nombres Bordx. 6, 301-325 (1994; Zbl 0827.11064)] in connection with the wild kernel of \(K\)-theory: as a matter of fact, when the number field \(K\) contains the \(2p\)th roots of unity for an odd prime \(p\), there exists a canonical isomorphism \(\mu_p \otimes\widetilde{\text{Cl}}_K\simeq WK_2(K)/ W(K_2(K)^p\) between the twisted logarithmic \(p\)-class group and the \(p\)-quotient of the wild kernel [Acta Arith. 67, 335-348 (1994; Zbl 0835.11042)]. And since the logarithmic class groups are easily computable, as explained by F. Diaz y Diaz and the reviewer [J. Number Theory 76, 1-15 (1999; Zbl 0930.11079)], this gives rise to an algorithmic approach to the Hilbert kernel. For \(p=2\) however, the isomorphism above is not valid without the assumption \(i\in K\) (in fact when \(i\) is not contained in the cyclotomic \(\mathbb{Z}_2\)-extension \(K^c\) of \(K\)) and the situation is more intricate.
In the article under review, the author defines a logarithmic signature that involves both the real places and certain 2-adic ones (namely the 2-adic places \({\mathfrak p}\) such that the cyclotomic \(\mathbb{Z}_2\)-extension \(K_{\mathfrak p}^c\) of the local field \(K_{\mathfrak p}\) does not contain the 4th root of unity \(i\)). By this way he extends a previous generalization of the reviewer [Manuscr. Math. 93, 409-420 (1997; Zbl 0887.11044)]. Unfortunately there is a misprint in Lemma 8 that affects some formulas in Section 3 of a factor equal to 1 or 2. This point will be corrected in a Corrigendum to appear. Finally applications to the wild kernel will also follow in a forthcoming article.


11R29 Class numbers, class groups, discriminants
11R70 \(K\)-theory of global fields
11R99 Algebraic number theory: global fields
Full Text: DOI Numdam EuDML


[1] Anglès, B., Jaulent, J.-F., Théorie des genres des corps globaux. Manuscripta Math.101 (2000), 513-532. · Zbl 0982.11062
[2] Federer, L.J., Gross, B.H., Regulators and Iwasawa modules. Inv. Math.62 (1981), 443-457. · Zbl 0468.12005
[3] Jaulent, J.-F., Classes logarithmiques des corps de nombres. J. Théor. Nombres Bordeaux6 (1994), 301-325. · Zbl 0827.11064
[4] Jaulent, J.-F., Théorie -adique du corps de classes. J. Théor. Nombres Bordeaux10 (1998), 355-397. · Zbl 0938.11052
[5] Jaulent, J.-F., Sauzet, O., Extensions quadratiques 2-birationnelles de corps de nombres totalement réels. Pub. Matemàtiques44 (2000), 343-353. · Zbl 0961.11037
[6] Soriano, F., Classes logarithmiques ambiges des corps quadratiques. Acta Arith.78 (1997), 201-219. · Zbl 0869.11081
[7] Soriano, F., Classes logarithmiques au sens restreint. Manuscripta Math.93 (1997), 409-420. · Zbl 0887.11044
[8] Soriano, F., Classes logarithmiques généralisées ambiges. Abh. Math. Sem. Univ. Hamburg68 (1998), 329-338. · Zbl 0953.11035
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.