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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
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