Global \(\ell\)-adique clean field theory. (Théorie \(\ell\)-adique globale du corps de classes.) (French) Zbl 0938.11052

The fundamental results of \(\ell\)-adic class field theory for number fields are presented here. The first section is devoted to the construction of the fundamental \(\mathbb{Z}_\ell\) module in the \(\ell\)-adic group of ideles. The fundamentals of \(\ell\)-adic class field theory are presented in the second section, while the third is devoted to the theory of \(\ell\)-adic duality. The following theorem of Section 2 helps to describe the general theory: Let \(K\) be a number field, then the reciprocity map induces a continuous isomorphism of the \(\ell\)-group \(J_K\) of ideles of \(K\) onto the Galois group \(G_K^{ab}= G(K^{ab}/K)\) of the maximal abelian \(\ell\)-extension \(K^{ab}\) of \(K\). The kernel of this mapping is the subgroup \(R_K\) of principal ideles. In this correspondence the decomposition subgroup \(D_\wp\) of a place \(\wp\) of \(K\) is the image of \(G_K^{ab}\) of the subgroup \(R_\wp\) of \(J_K\) and the inertia subgroup \(I_\wp\) is the image or the subgroup \(U_\wp\) of units of \(R_\wp\). Here \(R_\wp\) is the \(\ell\)-adic compactification of the multiplicative group \(K_\wp^*\) where \(K_\wp\) is the completion of \(K\) at a place \(\wp\) of \(K\).


11R37 Class field theory
11R56 Adèle rings and groups
11S37 Langlands-Weil conjectures, nonabelian class field theory
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[1] Artin, E., Tate, J., Class field theory, Benjamin, New York, 1967. · Zbl 0176.33504
[2] Bertrandias, F., Payan, J.-J., Γ-extensions et invariants cyclotomiques. Ann. Sci. Ec. Norm. Sup5 (1972), 517-548. · Zbl 0246.12005
[3] Federer, L.J., Gross, B.N., Regulators and Iwasawa modules. Inv. Math.62 (1981), 443-457. · Zbl 0468.12005
[4] Gillard, R., Formulations de la conjecture de Leopoldt et étude d’une condition suffisante. Abh. Math. Sem. Hamburg48 (1979), 125-138. · Zbl 0396.12008
[5] Gras, G., Jaulent, J.-F., Sur les corps de nombres réguliers. Math. Z.202 (1989), 343-365. · Zbl 0704.11040
[6] Gras, G., Plongements kummériens dans les Zp-extensions. Composito Math.55 (1985), 383-395. · Zbl 0584.12004
[7] Jaulent, J.-F., Nguyen Quang Do, T., Corps p-rationnels, corps p-réguliers et ramification restreinte. J. Théor. Nombres Bordeaux5 (1993), 343-363. · Zbl 0957.11046
[8] Jaulent, J.-F., L’arithmétique des l-extension (Thèse d’Etat), Pub. Math. Fac. Sci. Besançon Théor. Nombres 1985/86 (1986). · Zbl 0601.12002
[9] Jaulent, J.-F., Sur les conjectures de Leopoldt et de Gross. Journées arithmétiques de Besançon,Astérisque147-148 (1987), 107-120. · Zbl 0623.12003
[10] Jaulent, J.-F., Noyau universel et valeurs absolues Journées arithmétiques de Marseille-Luminy, Astérisque198-200 (1990), 187-209. · Zbl 0756.11033
[11] Jaulent, J.-F., La Théorie de Kummer et le K2 des corps de nombres. J. Théor. Nombres Bordeaux2 (1990), 377-411. · Zbl 0723.11051
[12] Jaulent, J.-F., Classes logarithmiques des corps de nombres. J. Théor. Nombres Bordeaux6 (1994), 301-325. · Zbl 0827.11064
[13] Jaulent, J.-F., Sauzet, O., Pro-l-extensions de corps de nombres Γ-rationnels. J. Numb. Th.65 (1997), 240-267. · Zbl 0896.11043
[14] Kolster, M., An idelic approach to the wild kernel. Invent. Math.103 (1991), 9-24. · Zbl 0724.11056
[15] Miki, H., On the Leopoldt conjecture on the p-adic regulators. J. Numb. Th.26 (1987), 117-128. · Zbl 0621.12009
[16] Miki, H., On the maximal abelian l-extension of a finite algebraic number field with given ramification. Nagoya Math. J.70 (1978), 183-202. · Zbl 0398.12003
[17] Movahhedi, A., Nguyen Quang Do, T., Sur l’arithmétique des corps de nombres p-rationnels. Sém. Th. Nombres Paris (1987/1988), Prog. in Math.89 (1990), 155-200. · Zbl 0703.11059
[18] Movahhedi, A., Sur les p-extensions des corps p-rationnels. Mat. Nachr.149 (1990), 163-176. · Zbl 0723.11054
[19] Nguyen Quang Do, T., Lois de réciprocité primitives. Manuscripta Math.72 (1991), 307-324. · Zbl 0747.11047
[20] Serre, J.-P., Cohomologie galoisienne. , Springer Verlag (1994). · Zbl 0812.12002
[21] Soriano, F., Classes logarithmiques ambiges des corps quadratiques. Acta Arith.78 (1997), 201-219. · Zbl 0869.11081
[22] Thomas, H., Premier étage d’une Zl-extension. Manuscripta Math.81 (1993), 413-435. · Zbl 0799.11047
[23] Thomas, H., Trivialité du 2-rang du noyau hilbertien. J. Théor. Nombres Bordeaux6 (1994), 459-483. · Zbl 0822.11079
[24] Tate, J., Les conjectures de Stark sur les fonctions L d’artin en s = 0. Prog. in Math.47, Birkhaüser, 1984. · Zbl 0545.12009
[25] Wingberg, K., On Galois groups of p-closed algebraic number fields with restricted ramification. J. reine angew. Math.400 (1989), 185-202. · Zbl 0715.11065
[26] Wingberg, K., On Galois groups of p-closed algebraic number fields with restricted ramification II. J. reine angew. Math.416 (1991), 187-194. · Zbl 0728.11058
[27] Yamagishi, M., A note on free pro-p-extensions of algebraic number fields. J. Théor. Nombres Bordeaux5 (1993), 165-178. · Zbl 0784.11052
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