The Idele Group

The invertible elements of any commutative topological ring $R$ are a group $R^*$ under multiplication. In general $R^*$ is not a topological group if it is endowed with the subset topology because inversion need not be continuous (only multiplication and addition on $R$ are required to be continuous). It is usual therefore to give $R^*$ the following topology. There is an injection

$$x\mapsto \left( x, \frac{1}{x} \right)$$ (19.1)

of $R^*$ into the topological product $R\times R$. We give $R^*$ the corresponding subset topology. Then $R^*$ with this topology is a topological group and the inclusion map $R^*\hookrightarrow R$ is continous. To see continuity of inclusion, note that this topology is finer (has at least as many open sets) than the subset topology induced by $R^*\subset R$, since the projection maps $R\times R\to R$ are continuous.

Example 19.1.1   This is a ``non-example''. The inverse map on $\mathbf{Z}_p^*$ is continuous with respect to the $p$-adic topology. If $a,b\in \mathbf{Z}_p^*$, then $\left\vert a\right\vert=\left\vert b\right\vert=1$, so if $\left\vert a-b\right\vert<\varepsilon$, then

$$\left\vert\frac{1}{a} - \frac{1}{b}\right\vert = \left\vert\frac{... ...a\right\vert}{\left\vert ab\right\vert} < \frac{\varepsilon }{1}=\varepsilon .$$

Definition 19.1.2 (Idele Group)   The idele group $\mathbb{I}_K$ of $K$ is the group $\AA _K^*$ of invertible elements of the adele ring $\AA _K$.

We shall usually speak of $\mathbb{I}_K$ as a subset of $\AA _K$, and will have to distinguish between the $\mathbb{I}_K$ and $\AA _K$-topologies.

Example 19.1.3   For a rational prime $p$, let $\mathbf{x}_p\in \AA_\mathbf{Q}$ be the adele whose $p$th component is $p$ and whose $v$th component, for $v\neq p$, is $1$. Then $\mathbf{x}_p \to 1$ as $p\to\infty$ in $\AA_\mathbf{Q}$, for the following reason. We must show that if $U$ is a basic open set that contains the adele $1=\{1\}_v$, the $\mathbf{x}_p$ for all sufficiently large $p$ are contained in $U$. Since $U$ contains $1$ and is a basic open set, it is of the form

$$\prod_{v\in S} U_v \times \prod_{v\not\in S} \mathbf{Z}_v,$$

where $S$ if a finite set, and the $U_v$, for $v\in S$, are arbitrary open subsets of $\mathbf{Q}_v$ that contain $1$. If $q$ is a prime larger than any prime in $S$, then $\mathbf{x}_p$ for $p\geq q$, is in $U$. This proves convergence. If the inverse map were continuous on $\mathbb{I}_K$, then the sequence of $\mathbf{x}_p^{-1}$ would converge to $1^{-1}=1$. However, if $U$ is an open set as above about $1$, then for sufficiently large $p$, none of the adeles $\mathbf{x}_p$ are contained in $U$.

Lemma 19.1.4   The group of ideles $\mathbb{I}_K$ is the restricted topological project of the $K_v^*$ with respect to the units $U_v=\O_v^*\subset K_v$, with the restricted product topology.

We omit the proof of Lemma 19.1.4, which is a matter of thinking carefully about the definitions. The main point is that inversion is continuous on $\O_v^*$ for each $v$. (See Example 19.1.1.)

We have seen that $K$ is naturally embedded in $\AA _K$, so $K^*$ is naturally embedded in  $\mathbb{I}_K$.

Definition 19.1.5 (Principal Ideles)   We call $K^*$, considered as a subgroup of $\mathbb{I}_K$, the principal ideles.

Lemma 19.1.6   The principal ideles $K^*$ are discrete as a subgroup of $\mathbb{I}_K$.

Proof. For $K$ is discrete in $\AA _K$, so $K^*$ is embedded in $\AA_K\times \AA_K$ by (19.1.1) as a discrete subset. (Alternatively, the subgroup topology on $\mathbb{I}_K$ is finer than the topology coming from $\mathbb{I}_K$ being a subset of $\AA _K$, and $K$ is already discrete in $\AA _K$.) $\qedsymbol$

Definition 19.1.7 (Content of an Idele)   The content of $\mathbf{x}=\{x_v\}_v \in \mathbb{I}_K$ is

$$c(\mathbf{x}) = \prod_{\text{all }v} \left\vert x_v\right\vert _v \in \mathbf{R}_{>0}.$$

Lemma 19.1.8   The map $\mathbf{x}\to c(\mathbf{x})$ is a continuous homomorphism of the topological group $\mathbb{I}_K$ into $\mathbf{R}_{>0}$, where we view $\mathbf{R}_{>0}$ as a topological group under multiplication. If $K$ is a number field, then $c$ is surjective.

Proof. That the content map $c$ satisfies the axioms of a homomorphisms follows from the multiplicative nature of the defining formula for $c$. For continuity, suppose $(a,b)$ is an open interval in $\mathbf{R}_{>0}$. Suppose $\mathbf{x}\in\mathbb{I}_K$ is such that $c(\mathbf{x})\in (a,b)$. By considering small intervals about each non-unit component of $\mathbf{x}$, we find an open neighborhood $U\subset \mathbb{I}_K$ of $\mathbf{x}$ such that $c(U)\subset (a,b)$. It follows the $c^{-1}((a,b))$ is open.

For surjectivity, use that each archimedean valuation is surjective, and choose an idele that is $1$ at all but one archimedean valuation. $\qedsymbol$

Remark 19.1.9   Note also that the $\mathbb{I}_K$-topology is that appropriate to a group of operators on $\AA _K^+$: a basis of open sets is the $S(C,U)$, where $C,U\subset \AA_K^+$ are, respectively, $\AA _K$-compact and $\AA _K$-open, and $S$ consists of the $\mathbf{x}\in \mathbb{I}_J$ such that $(1-\mathbf{x}) C\subset U$ and $(1-\mathbf{x}^{-1}) C\subset U$.

Definition 19.1.10 ($1$-Ideles)   The subgroup $\mathbb{I}_K^1$ of $1$-ideles is the subgroup of ideles $\mathbf{x}=\{x_v\}$ such that $c(\mathbf{x})=1$. Thus $\mathbb{I}_K^1$ is the kernel of $c$, so we have an exact sequence

$$1 \to \mathbb{I}_K^1 \to \mathbb{I}_K \xrightarrow{c} \mathbf{R}_{>0}\to 1,$$

where the surjectivity on the right is only if $K$ is a number field.

Lemma 19.1.11   The subset $\mathbb{I}_K^1$ of $\AA _K$ is closed as a subset, and the $\AA _K$-subset topology on $\mathbb{I}_K^1$ coincides with the $\mathbb{I}_K$-subset topology on $\mathbb{I}_K^1$.

Proof. Let $\mathbf{x}\in \AA _K$ with $\mathbf{x}\not\in\mathbb{I}_K^1$. To prove that $\mathbb{I}_K^1$ is closed in $\AA _K$, we find an $\AA _K$-neighborhood $W$ of $\mathbf{x}$ that does not meet $\mathbb{I}_K^1$.

1st Case. Suppose that $\prod_v \left\vert x_v\right\vert _v<1$ (possibly $=0$). Then there is a finite set $S$ of $v$ such that

1. $S$ contains all the $v$ with $\left\vert x_v\right\vert _v>1$, and
2. $\prod_{v\in S}\left\vert x_v\right\vert _v < 1$.

Then the set $W$ can be defined by

$$\left\vert w_v - x_v\right\vert _v < \varepsilon \qquad v\in S$$

$$\left\vert w_v\right\vert _v \leq 1 \qquad v\not\in S$$

for sufficiently small $\varepsilon$.

2nd Case. Suppose that $C:=\prod_v \left\vert x_v\right\vert _v>1$. Then there is a finite set $S$ of $v$ such that

1. $S$ contains all the $v$ with $\left\vert x_v\right\vert _v>1$, and
2. if $v\not\in S$ an inequality $\left\vert w_v\right\vert _v<1$ implies $\left\vert w_v\right\vert _v < \frac{1}{2C}.$ (This is because for a non-archimedean valuation, the largest absolute value less than $1$ is $1/p$, where $p$ is the residue characteristic. Also, the upper bound in Cassels's article is $\frac{1}{2}C$ instead of $\frac{1}{2C}$, but I think he got it wrong.)

We can choose $\varepsilon$ so small that $\left\vert w_v-x_v\right\vert _v<\varepsilon$ (for $v\in S$) implies $1<\prod_{v\in S} \left\vert w_v\right\vert _v < 2C.$ Then $W$ may be defined by

$$\left\vert w_v - x_v\right\vert _v < \varepsilon \qquad v\in S$$

$$\left\vert w_v\right\vert _v \leq 1 \qquad v\not\in S.$$

This works because if $\mathbf{w}\in W$, then either $\left\vert w_v\right\vert _v=1$ for all $v\not\in S$, in which case $1 < c(\mathbf{w}) < 2c$, so $\mathbf{w}\not\in \mathbb{I}_K^1$, or $\left\vert w_{v_0}\right\vert _{v_0}<1$ for some $v_0\not\in S$, in which case

$$c(\mathbf{w}) = \left(\prod_{v\in S} \left\vert w_v\right\vert _v... ...\cdot \left\vert w_{v_0}\right\vert \cdots < 2C \cdot \frac{1}{2C} \cdots < 1,$$

so again $\mathbf{w}\not\in \mathbb{I}_K^1$.

We next show that the $\mathbb{I}_K$- and $\AA _K$-topologies on $\mathbb{I}_K^1$ are the same. If $\mathbf{x}\in \mathbb{I}_K^1$, we must show that every $\AA _K$-neighborhood of $\mathbf{x}$ contains an $\AA _K$-neighborhood and vice-versa.

Let $W\subset \mathbb{I}_K^1$ be an $\AA _K$-neighborhood of $\mathbf{x}$. Then it contains an $\AA _K$-neighborhood of the type

$$\left\vert w_v - x_v\right\vert _v < \varepsilon \qquad v\in S$$ (19.2)

$$\left\vert w_v\right\vert _v \leq 1 \qquad v\not\in S$$ (19.3)

where $S$ is a finite set of valuations $v$. This contains the $\mathbb{I}_K$-neighborhood in which $\leq$ in (19.1.2) is replaced by $=$.

Next let $H\subset \mathbb{I}_K^1$ be an $\mathbb{I}_K$-neighborhood. Then it contains an $\mathbb{I}_K$-neighborhood of the form

$$\left\vert w_v - x_v\right\vert _v < \varepsilon \qquad v\in S$$ (19.4)

$$\left\vert w_v\right\vert _v = 1 \qquad v\not\in S,$$ (19.5)

where the finite set $S$ contains at least all archimedean valuations $v$ and all valuations $v$ with $\left\vert x_v\right\vert _v\neq 1$. Since $\prod\left\vert x_v\right\vert _v=1$, we may also suppose that $\varepsilon$ is so small that (19.1.4) implies

$$\prod_v\left\vert w_v\right\vert _v < 2.$$

Then the intersection of (19.1.4) with $\mathbb{I}_K^1$ is the same as that of (19.1.2) with $\mathbb{I}_K^1$, i.e., (19.1.4) defines an $\AA _K$-neighborhood. $\qedsymbol$

By the product formula we have that $K^*\subset \mathbb{I}_K^1$. The following result is of vital importance in class field theory.

Theorem 19.1.12   The quotient $\mathbb{I}_K^1/K^*$ with the quotient topology is compact.

Proof. After the preceeding lemma, it is enough to find an $\AA _K$-compact set $W\subset \AA _K$ such that the map

$$W\cap \mathbb{I}_K^1 \to \mathbb{I}_K^1/K^*$$

is surjective. We take for $W$ the set of $\mathbf{w}=\{w_v\}_v$ with

$$\left\vert w_v\right\vert _v\leq \left\vert x_v\right\vert _v,$$

where $\mathbf{x}=\{x_v\}_v$ is any idele of content greater than the $C$ of Lemma 18.4.1.

Let $\mathbf{y}=\{y_v\}_v\in \mathbb{I}_K^1$. Then the content of $\mathbf{x}/\mathbf{y}$ equals the content of $\mathbf{x}$, so by Lemma 18.4.1 there is an $a\in K^*$ such that

$$\left\vert a\right\vert _v \leq \left\vert\frac{x_v}{y_v}\right\vert _v$$   all $v$$$.$$

Then $a\mathbf{y}\in W$, as required. $\qedsymbol$

Remark 19.1.13   The quotient $\mathbb{I}_K^1/K^*$ is totally disconnected in the function field case. For the structure of its connected component in the number field case, see papers of Artin and Weil in the ``Proceedings of the Tokyo Symposium on Algebraic Number Theory, 1955'' (Science Council of Japan) or [AT90]. The determination of the character group of $\mathbb{I}_K/K^*$ is global class field theory.