idèle

Let $K$ be a number field. For each finite prime $v$ of $K$ , let $\mathfrak{o}_{v}$ be the valuation ring of the completion $K_{v}$ of $K$ at $v$ , and let $U_{v}$ be the group of units in $\mathfrak{o}_{v}$ . Then each group $U_{v}$ is a compact open subgroup of the group of units $K_{v}^{*}$ of $K_{v}$ . The idèle group $\mathbb{I}_{K}$ of $K$ is defined to be the restricted direct product of the multiplicative groups $\{K_{v}^{*}\}$ with respect to the compact open subgroups $\{U_{v}\}$ , taken over all finite primes and infinite primes $v$ of $K$ .

The units $K^{*}$ in $K$ embed into $\mathbb{I}_{K}$ via the diagonal embedding

$x\mapsto\prod_{v}x_{v},$

where $x_{v}$ is the image of $x$ under the embedding $K\hookrightarrow K_{v}$ of $K$ into its completion $K_{v}$ . As in the case of adèles, the group $K^{*}$ is a discrete subgroup of the group of idèles $\mathbb{I}_{K}$ , but unlike the case of adèles, the quotient group $\mathbb{I}_{K}/K^{*}$ is not a compact group. It is, however, possible to define a certain subgroup of the idèles (the subgroup of norm 1 elements) which does have compact quotient under $K^{*}$ .

Warning: The group $\mathbb{I}_{K}$ is a multiplicative subgroup of the ring of adèles $\mathbb{A}_{K}$ , but the topology on $\mathbb{I}_{K}$ is different from the subspace topology that $\mathbb{I}_{K}$ would have as a subset of $\mathbb{A}_{K}$ .

Title	idèle
Canonical name	Idele
Date of creation	2013-03-22 12:39:28
Last modified on	2013-03-22 12:39:28
Owner	djao (24)
Last modified by	djao (24)
Numerical id	7
Author	djao (24)
Entry type	Definition
Classification	msc 11R56
Related topic	Adele
Defines	idèle group
Defines	group of idèles