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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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