Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0938.11052
Jaulent, Jean-François
Global $\ell$-adique clean field theory. (Théorie $\ell$-adique globale du corps de classes.) (French)
[J] J. Théor. Nombres Bordx. 10, No.2, 355-397 (1998). ISSN 1246-7405

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 $\bbfZ_\ell$ 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 $K$.
[C.Parry (Blacksburg)]

MSC 2000:: *11R37 Class field theory for global fields
11R56 Adele rings and groups
11S37 Langlands-Weil conjectures, nonabelian class field theory

Keywords: $\ell$-adic class field theory; $\ell$-adic group of ideles; $\ell$-adic duality; reciprocity map; ideles

Cited in: Zbl 1163.11071

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Article Article Journal Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]