Adele ring

In mathematics, the adele ring (also adelic ring, ring of adeles or ring of adèles) is defined in class field theory, a branch of algebraic number theory. It allows one to elegantly describe the Artin reciprocity law. The adele ring is a self-dual topological ring, which is built on a global field. It is the restricted product of all the completions of the global field and therefore contains all the completions of the global field.

The idele class group, which is the quotient group of the group of units of the adele ring by the group of units of the global field, is a central object in class field theory.

Origin of the name

In local class field theory, the group of units of the local field plays a central role. In global class field theory, the idele class group takes this role. The term "idele" (French: idèle) is an invention of the French mathematician Claude Chevalley (1909–1984) and stands for "ideal element" (abbreviated: id.el.). The term "adele" (adèle) stands for additive idele.

The idea of the adele ring is to look at all completions of $K$ at once. At first glance, the Cartesian product could be a good candidate. However, the adele ring is defined with the restricted product. There are two reasons for this:

• For each element of $K$ the valuations are zero for almost all places, i.e., for all places except a finite number. So, the global field can be embedded in the restricted product.

• The restricted product is a locally compact space, while the Cartesian product is not. Therefore, we can't apply harmonic analysis to the Cartesian product.

Notation and basic definitions

Global fields

Throughout this article, $K$ is a global field, meaning it is either a number field (a finite extension of $\mathbb {Q}$) or a global function field (a finite extension of $\mathbb {F} _{p^{r}}(t)$ for $p$ prime and $r\in \mathbb {N}$). By definition a finite extension of a global field is itself a global field.

Valuations

For a valuation $v$ of $K$ we write $K_{v}$ for the completion of $K$ with respect to $v.$ If $v$ is discrete we write $O_{v}$ for the valuation ring of $K_{v}$ and ${\mathfrak {m}}_{v}$ for the maximal ideal of $O_{v}.$ If this is a principal ideal we denote the uniformizing element by $\pi _{v}.$ A non-Archimedean valuation is written as $v<\infty$ or $v\nmid \infty$ and an Archimedean valuation as $v|\infty .$ We assume all valuations to be non-trivial.

There is a one-to-one identification of valuations and absolute values. Fix a constant $C>1,$ the valuation $v$ is assigned the absolute value $|\cdot |_{v},$ defined as:

$\forall x\in K:\quad |x|_{v}:={\begin{cases}C^{-v(x)}&x\neq 0\\0&x=0\end{cases}}$

Conversely, the absolute value $|\cdot |$ is assigned the valuation $v_{|\cdot |},$ defined as:

$\forall x\in K^{\times }:\quad v_{|\cdot |}(x):=-\log _{C}(|x|).$

A place of $K$ is a representative of an equivalence class of valuations (or absolute values) of $K.$ Places corresponding to non-Archimedean valuations are called finite, whereas places corresponding to Archimedean valuations are called infinite. The set of infinite places of a global field is finite, we denote this set by $P_{\infty }.$

Define $\textstyle {\widehat {O}}:=\prod _{v<\infty }O_{v}$ and let ${\widehat {O}}^{\times }$ be its group of units. Then $\textstyle {\widehat {O}}^{\times }=\prod _{v<\infty }O_{v}^{\times }.$

Finite extensions

Let $L/K$ be a finite extension of the global field $K.$ Let $w$ be a place of $L$ and $v$ a place of $K.$ We say $w$ lies above $v,$ denoted by $w|v,$ if the absolute value $|\cdot |_{w}$ restricted to $K$ is in the equivalence class of $v.$ Define

${\begin{aligned}L_{v}&:=\prod _{w|v}L_{w},\\{\widetilde {O_{v}}}&:=\prod _{w|v}O_{w}.\end{aligned}}$

Note that both products are finite.

If $w|v$ we can embed $K_{v}$ in $L_{w}.$ Therefore, we can embed $K_{v}$ diagonally in $L_{v}.$ With this embedding $L_{v}$ is a commutative algebra over $K_{v}$ with degree

$\sum _{w|v}[L_{w}:K_{v}]=[L:K].$

The adele ring

The set of finite adeles of a global field $K,$ denoted $\mathbb {A} _{K,{\text{fin}}},$ is defined as the restricted product of $K_{v}$ with respect to the $O_{v}:$

$\mathbb {A} _{K,{\text{fin}}}:={\prod _{v<\infty }}^{'}K_{v}=\left\{\left.(x_{v})_{v}\in \prod _{v<\infty }K_{v}\right|x_{v}\in O_{v}{\text{ for almost all }}v\right\}.$

It is equipped with the restricted product topology, the topology generated by restricted open rectangles, which have the following form:

$U=\prod _{v\in E}U_{v}\times \prod _{v\notin E}O_{v}\subset {\prod _{v<\infty }}^{'}K_{v},$

where $E$ is a finite set of places and $U_{v}\subset K_{v}$ are open. With component-wise addition and multiplication $\mathbb {A} _{K,{\text{fin}}}$ is also a ring.

The adele ring of a global field $K$ is defined as the product of $\mathbb {A} _{K,{\text{fin}}}$ with the product of the completions of $K$ at its infinite places. The number of infinite places is finite and the completions are either $\mathbb {R}$ or $\mathbb {C} .$ In short:

$\mathbb {A} _{K}:=\mathbb {A} _{K,{\text{fin}}}\times \prod _{v|\infty }K_{v}={\prod _{v<\infty }}^{'}K_{v}\times \prod _{v|\infty }K_{v}.$

With addition and multiplication defined as component-wise the adele ring is a ring. The elements of the adele ring are called adeles of $K.$ In the following, we write

$\mathbb {A} _{K}={\prod _{v}}^{'}K_{v},$

although this is generally not a restricted product.

Remark. Global function fields do not have any infinite places and therefore the finite adele ring equals the adele ring.

Lemma. There is a natural embedding of $K$ into $\mathbb {A} _{K}$ given by the diagonal map: $a\mapsto (a,a,\ldots ).$

Proof. If $a\in K,$ then $a\in O_{v}^{\times }$ for almost all $v.$ This shows the map is well-defined. It is also injective because the embedding of $K$ in $K_{v}$ is injective for all $v.$

Remark. By identifying $K$ with its image under the diagonal map we regard it as a subring of $\mathbb {A} _{K}.$ The elements of $K$ are called the principal adeles of $\mathbb {A} _{K}.$

Definition. Let $S$ be a set of places of $K.$ Define the set of the $S$-adeles of $K$ as

$\mathbb {A} _{K,S}:={\prod _{v\in S}}^{'}K_{v}.$

Furthermore if we define

$\mathbb {A} _{K}^{S}:={\prod _{v\notin S}}^{'}K_{v}$

we have: $\mathbb {A} _{K}=\mathbb {A} _{K,S}\times \mathbb {A} _{K}^{S}.$

The adele ring of rationals

By Ostrowski's theorem the places of $\mathbb {Q}$ are $\{p\in \mathbb {N} :p{\text{ prime}}\}\cup \{\infty \},$ where we identify a prime $p$ with the equivalence class of the $p$-adic absolute value and $\infty$ with the equivalence class of the absolute value $|\cdot |_{\infty }$ defined as:

$\forall x\in \mathbb {Q} :\quad |x|_{\infty }:={\begin{cases}x&x\geq 0\\-x&x<0\end{cases}}$

The completion of $\mathbb {Q}$ with respect to the place $p$ is $\mathbb {Q} _{p}$ with valuation ring $\mathbb {Z} _{p}.$ For the place $\infty$ the completion is $\mathbb {R} .$ Thus:

${\begin{aligned}\mathbb {A} _{\mathbb {Q} ,{\text{fin}}}&={\prod _{p<\infty }}^{'}\mathbb {Q} _{p}\\\mathbb {A} _{\mathbb {Q} }&=\left({\prod _{p<\infty }}^{'}\mathbb {Q} _{p}\right)\times \mathbb {R} \end{aligned}}$

Or for short

$\mathbb {A} _{\mathbb {Q} }={\prod _{p\leq \infty }}^{'}\mathbb {Q} _{p},\qquad \mathbb {Q} _{\infty }:=\mathbb {R} .$

We will illustrate the difference between restricted and unrestricted product topology using a sequence in $\mathbb {A} _{\mathbb {Q} }$:

Lemma. Consider the following sequence in $\mathbb {A} _{\mathbb {Q} }$:

${\begin{aligned}x_{1}&=\left({\frac {1}{2}},1,1,\ldots \right)\\x_{2}&=\left(1,{\frac {1}{3}},1,\ldots \right)\\x_{3}&=\left(1,1,{\frac {1}{5}},1,\ldots \right)\\x_{4}&=\left(1,1,1,{\frac {1}{7}},1,\ldots \right)\\&\vdots \end{aligned}}$

In product topology it converges to $(1,1,\ldots ).$ It doesn't converge in restricted product topology.

Proof. In product topology convergence corresponds to the convergence in each coordinate, which is trivial because the sequences become stationary. The sequence doesn't converge in restricted product topology, for each adele $a=(a_{p})_{p}\in \mathbb {A} _{\mathbb {Q} }$ and for each restricted open rectangle $\textstyle U=\prod _{p\in E}U_{p}\times \prod _{p\notin E}\mathbb {Z} _{p},$ we have: ${\tfrac {1}{p}}-a_{p}\notin \mathbb {Z} _{p}$ for $a_{p}\in \mathbb {Z} _{p}$ and therefore ${\tfrac {1}{p}}-a_{p}\notin \mathbb {Z} _{p}$ for all $p\notin F.$ As a result $x_{n}-a\notin U$ for almost all $n\in \mathbb {N} .$ In this consideration, $E$ and $F$ are finite subsets of the set of all places.

Alternative definition for number fields

Definition (profinite integers). We define profinite integers as the profinite completion of the rings $\mathbb {Z} /n\mathbb {Z}$ with the partial order $n\geq m\Leftrightarrow m|n,$ i.e.,

${\widehat {\mathbb {Z} }}:=\varprojlim _{n}\mathbb {Z} /n\mathbb {Z} ,$

Lemma. $\textstyle {\widehat {\mathbb {Z} }}\cong \prod _{p}\mathbb {Z} _{p}.$

Proof. This follows from Chinese Remainder Theorem.

Lemma. $\mathbb {A} _{\mathbb {Q} ,{\text{fin}}}={\widehat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q} .$

Proof. We will use the universal property of the tensor product. Define a $\mathbb {Z}$-bilinear function

${\begin{cases}\Psi :{\widehat {\mathbb {Z} }}\times \mathbb {Q} \to \mathbb {A} _{\mathbb {Q} ,{\text{fin}}}\\\left((a_{p})_{p},q\right)\mapsto (a_{p}q)_{p}\end{cases}}$

This is well-defined because for a given $q={\tfrac {m}{n}}\in \mathbb {Q}$ with $m,n$ co-prime there are only finitely many primes dividing $n.$ Let $M$ be another $\mathbb {Z}$-module with a $\mathbb {Z}$-bilinear map $\Phi :{\widehat {\mathbb {Z} }}\times \mathbb {Q} \to M.$ We have to show $\Phi$ factors through $\Psi$ uniquely, i.e., there exists a unique $\mathbb {Z}$-linear map ${\tilde {\Phi }}:\mathbb {A} _{\mathbb {Q} ,{\text{fin}}}\to M$ such that $\Phi ={\tilde {\Phi }}\circ \Psi .$ We define ${\tilde {\Phi }}$ as follows: for a given $(u_{p})_{p}$ there exist $u\in \mathbb {N}$ and $(v_{p})_{p}\in {\widehat {\mathbb {Z} }}$ such that $u_{p}={\tfrac {1}{u}}\cdot v_{p}$ for all $p.$ Define ${\tilde {\Phi }}((u_{p})_{p}):=\Phi ((v_{p})_{p},{\tfrac {1}{u}}).$ One can show ${\tilde {\Phi }}$ is well-defined, $\mathbb {Z}$-linear, satisfies $\Phi ={\tilde {\Phi }}\circ \Psi$ and is unique with these properties.

Corollary. Define $\mathbb {A} _{\mathbb {Z} }:={\widehat {\mathbb {Z} }}\times \mathbb {R} .$ Then we have an algebraic isomorphism $\mathbb {A} _{\mathbb {Q} }\cong \mathbb {A} _{\mathbb {Z} }\otimes _{\mathbb {Z} }\mathbb {Q} .$

Proof. $\mathbb {A} _{\mathbb {Z} }\otimes _{\mathbb {Z} }\mathbb {Q} =\left({\widehat {\mathbb {Z} }}\times \mathbb {R} \right)\otimes _{\mathbb {Z} }\mathbb {Q} \cong \left({\widehat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q} \right)\times (\mathbb {R} \otimes _{\mathbb {Z} }\mathbb {Q} )\cong \left({\widehat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q} \right)\times \mathbb {R} =\mathbb {A} _{\mathbb {Q} ,{\text{fin}}}\times \mathbb {R} =\mathbb {A} _{\mathbb {Q} }.$

Lemma. For a number field $K,\mathbb {A} _{K}=\mathbb {A} _{\mathbb {Q} }\otimes _{\mathbb {Q} }K.$

Remark. Using $\mathbb {A} _{\mathbb {Q} }\otimes _{\mathbb {Q} }K\cong \mathbb {A} _{\mathbb {Q} }\oplus \dots \oplus \mathbb {A} _{\mathbb {Q} },$ where there are $[K:\mathbb {Q} ]$ summands, we give the right side the product topology and transport this topology via the isomorphism onto $\mathbb {A} _{\mathbb {Q} }\otimes _{\mathbb {Q} }K.$

The adele ring of a finite extension

If $L/K$ be a finite extension then $L$ is a global field and thus $\mathbb {A} _{L}$ is defined and $\textstyle \mathbb {A} _{L}={\prod _{v}}^{'}L_{v}.$ We claim $\mathbb {A} _{K}$ can be identified with a subgroup of $\mathbb {A} _{L}.$ Map $a=(a_{v})_{v}\in \mathbb {A} _{K}$ to $a'=(a'_{w})_{w}\in \mathbb {A} _{L}$ where $a'_{w}=a_{v}\in K_{v}\subset L_{w}$ for $w|v.$ Then $a=(a_{w})_{w}\in \mathbb {A} _{L}$ is in the subgroup $\mathbb {A} _{K},$ if $a_{w}\in K_{v}$ for $w|v$ and $a_{w}=a_{w'}$ for all $w,w'$ lying above the same place $v$ of $K.$

Lemma. If $L/K$ is a finite extension then $\mathbb {A} _{L}\cong \mathbb {A} _{K}\otimes _{K}L$ both algebraically and topologically.

With the help of this isomorphism, the inclusion $\mathbb {A} _{K}\subset \mathbb {A} _{L}$ is given by

${\begin{cases}\mathbb {A} _{K}\to \mathbb {A} _{L}\\\alpha \mapsto \alpha \otimes _{K}1\end{cases}}$

Furthermore, the principal adeles in $\mathbb {A} _{K}$ can be identified with a subgroup of principal adeles in $\mathbb {A} _{L}$ via the map

${\begin{cases}K\to (K\otimes _{K}L)\cong L\\\alpha \mapsto 1\otimes _{K}\alpha \end{cases}}$

Proof. Let $\omega _{1},\ldots ,\omega _{n}$ be a basis of $L$ over $K.$ Then for almost all $v,$

${\widetilde {O_{v}}}\cong O_{v}\omega _{1}\oplus \cdots \oplus O_{v}\omega _{n}.$

Furthermore, there are the following isomorphisms:

$K_{v}\omega _{1}\oplus \cdots \oplus K_{v}\omega _{n}\cong K_{v}\otimes _{K}L\cong L_{v}=\prod \nolimits _{w|v}L_{w}$

For the second we used the map:

${\begin{cases}K_{v}\otimes _{K}L\to L_{v}\\\alpha _{v}\otimes a\mapsto (\alpha _{v}\cdot (\tau _{w}(a)))_{w}\end{cases}}$

in which $\tau _{w}:L\to L_{w}$ is the canonical embedding and $w|v.$ We take on both sides the restricted product with respect to ${\widetilde {O_{v}}}:$

${\begin{aligned}\mathbb {A} _{K}\otimes _{K}L&=\left({\prod _{v}}^{'}K_{v}\right)\otimes _{K}L\\&\cong {\prod _{v}}^{'}(K_{v}\omega _{1}\oplus \cdots \oplus K_{v}\omega _{n})\\&\cong {\prod _{v}}^{'}(K_{v}\otimes _{K}L)\\&\cong {\prod _{v}}^{'}L_{v}\\&=\mathbb {A} _{L}\end{aligned}}$

Corollary. As additive groups $\mathbb {A} _{L}\cong \mathbb {A} _{K}\oplus \cdots \oplus \mathbb {A} _{K},$ where the left side has $[L:K]$ summands.

The set of principal adeles in $\mathbb {A} _{L}$ is identified with the set $K\oplus \cdots \oplus K,$ where the left side has $[L:K]$ summands and we consider $K$ as a subset of $\mathbb {A} _{K}.$

The adele ring of vector-spaces and algebras

Lemma. Suppose $P\supset P_{\infty }$ is a finite set of places of $K$ and define

$\mathbb {A} _{K}(P):=\prod _{v\in P}K_{v}\times \prod _{v\notin P}O_{v}.$

Equip $\mathbb {A} _{K}(P)$ with the product topology and define addition and multiplication component-wise. Then $\mathbb {A} _{K}(P)$ is a locally compact topological ring.

Remark. If $P'$ is another finite set of places of $K$ containing $P$ then $\mathbb {A} _{K}(P)$ is an open subring of $\mathbb {A} _{K}(P').$

Now, we are able to give an alternative characterization of the adele ring. The adele ring is the union of all sets $\mathbb {A} _{K}(P)$:

$\mathbb {A} _{K}=\bigcup _{P\supset P_{\infty },|P|<\infty }\mathbb {A} _{K}(P).$

Equivalently $\mathbb {A} _{K}$ is the set of all $x=(x_{v})_{v}$ so that $|x_{v}|_{v}\leq 1$ for almost all $v<\infty .$ The topology of $\mathbb {A} _{K}$ is induced by the requirement that all $\mathbb {A} _{K}(P)$ be open subrings of $\mathbb {A} _{K}.$ Thus, $\mathbb {A} _{K}$ is a locally compact topological ring.

Fix a place $v$ of $K.$ Let $P$ be a finite set of places of $K,$ containing $v$ and $P_{\infty }.$ Define

$\mathbb {A} _{K}'(P,v):=\prod _{w\in P\setminus \{v\}}K_{w}\times \prod _{w\notin P}O_{w}.$

Then:

$\mathbb {A} _{K}(P)\cong K_{v}\times \mathbb {A} _{K}'(P,v).$

Furthermore, define

$\mathbb {A} _{K}'(v):=\bigcup _{P\supset P_{\infty }\cup \{v\}}\mathbb {A} _{K}'(P,v),$

where $P$ runs through all finite sets containing $P_{\infty }\cup \{v\}.$ Then:

$\mathbb {A} _{K}\cong K_{v}\times \mathbb {A} _{K}'(v),$

via the map $(a_{w})_{w}\mapsto (a_{v},(a_{w})_{w\neq v}).$ The entire procedure above holds with a finite subset ${\widetilde {P}}$ instead of $\{v\}.$

By construction of $\mathbb {A} _{K}'(v),$ there is a natural embedding: $K_{v}\hookrightarrow \mathbb {A} _{K}.$ Furthermore, there exists a natural projection $\mathbb {A} _{K}\twoheadrightarrow K_{v}.$

The adele ring of a vector-space

Let $E$ be a finite dimensional vector-space over $K$ and $\{\omega _{1},\ldots ,\omega _{n}\}$ a basis for $E$ over $K.$ For each place $v$ of $K,$ we write:

${\begin{aligned}E_{v}&:=E\otimes _{K}K_{v}\cong K_{v}\omega _{1}\oplus \cdots \oplus K_{v}\omega _{n}\\{\widetilde {O_{v}}}&:=O_{v}\omega _{1}\oplus \cdots \oplus O_{v}\omega _{n}\end{aligned}}$

We define the adele ring of $E$ as

$\mathbb {A} _{E}:={\prod _{v}}^{'}E_{v}.$

This definition is based on the alternative description of the adele ring as a tensor product equipped with the same topology we defined when giving an alternate definition of adele ring for number fields. We equip $\mathbb {A} _{E}$ with the restricted product topology. Then $\mathbb {A} _{E}=E\otimes _{K}\mathbb {A} _{K}$ and we can embed $E$ in $\mathbb {A} _{E}$ naturally via the map $e\mapsto e\otimes 1.$

We give an alternative definition of the topology on $\mathbb {A} _{E}.$ Consider all linear maps: $E\to K.$ Using the natural embeddings $E\to \mathbb {A} _{E}$ and $K\to \mathbb {A} _{K},$ extend these linear maps to: $\mathbb {A} _{E}\to \mathbb {A} _{K}.$ The topology on $\mathbb {A} _{E}$ is the coarsest topology for which all these extensions are continuous.

We can define the topology in a different way. Fixing a basis for $E$ over $K$ results in an isomorphism $E\cong K^{n}.$ Therefore fixing a basis induces an isomorphism $(\mathbb {A} _{K})^{n}\cong \mathbb {A} _{E}.$ We supply the left hand side with the product topology and transport this topology with the isomorphism onto the right hand side. The topology doesn't depend on the choice of the basis, because another basis defines a second isomorphism. By composing both isomorphisms, we obtain a linear homeomorphism which transfers the two topologies into each other. More formally

${\begin{aligned}\mathbb {A} _{E}&=E\otimes _{K}\mathbb {A} _{K}\\&\cong (K\otimes _{K}\mathbb {A} _{K})\oplus \cdots \oplus (K\otimes _{K}\mathbb {A} _{K})\\&\cong \mathbb {A} _{K}\oplus \cdots \oplus \mathbb {A} _{K}\end{aligned}}$

where the sums have $n$ summands. In case of $E=L,$ the definition above is consistent with the results about the adele ring of a finite extension $L/K.$

The adele ring of an algebra

Let $A$ be a finite-dimensional algebra over $K.$ In particular, $A$ is a finite-dimensional vector-space over $K.$ As a consequence, $\mathbb {A} _{A}$ is defined and $\mathbb {A} _{A}\cong \mathbb {A} _{K}\otimes _{K}A.$ Since we have a multiplication on $\mathbb {A} _{K}$ and $A,$ we can define a multiplication on $\mathbb {A} _{A}$ via:

$\forall \alpha ,\beta \in \mathbb {A} _{K}{\text{ and }}\forall a,b\in A:\qquad (\alpha \otimes _{K}a)\cdot (\beta \otimes _{K}b):=(\alpha \beta )\otimes _{K}(ab).$

As a consequence, $\mathbb {A} _{A}$ is an algebra with a unit over $\mathbb {A} _{K}.$ Let ${\mathcal {B}}$ be a finite subset of $A,$ containing a basis for $A$ over $K.$ For any finite place $v$ we define $M_{v}$ as the $O_{v}$-module generated by ${\mathcal {B}}$ in $A_{v}.$ For each finite set of places, $P\supset P_{\infty },$ we define

$\mathbb {A} _{A}(P,\alpha )=\prod _{v\in P}A_{v}\times \prod _{v\notin P}M_{v}.$

One can show there is a finite set $P_{0},$ so that $\mathbb {A} _{A}(P,\alpha )$ is an open subring of $\mathbb {A} _{A},$ if $P\supset P_{0}.$ Furthermore $\mathbb {A} _{A}$ is the union of all these subrings and for $A=K,$ the definition above is consistent with the definition of the adele ring.

Trace and norm on the adele ring

Let $L/K$ be a finite extension. Since $\mathbb {A} _{K}=\mathbb {A} _{K}\otimes _{K}K$ and $\mathbb {A} _{L}=\mathbb {A} _{K}\otimes _{K}L$ from Lemma above we can interpret $\mathbb {A} _{K}$ as a closed subring of $\mathbb {A} _{L}.$ We write $\operatorname {con} _{L/K}$ for this embedding. Explicitly for all places $w$ of $L$ above $v$ and for any $\alpha \in \mathbb {A} _{K},(\operatorname {con} _{L/K}(\alpha ))_{w}=\alpha _{v}\in K_{v}.$

Let $M/L/K$ be a tower of global fields. Then:

$\operatorname {con} _{M/K}(\alpha )=\operatorname {con} _{M/L}(\operatorname {con} _{L/K}(\alpha ))\qquad \forall \alpha \in \mathbb {A} _{K}.$

Furthermore, restricted to the principal adeles $\operatorname {con}$ is the natural injection $K\to L.$

Let $\{\omega _{1},\ldots ,\omega _{n}\}$ be a basis of the field extension $L/K.$ Then each $\alpha \in \mathbb {A} _{L}$ can be written as $\textstyle \sum _{j=1}^{n}\alpha _{j}\omega _{j},$ where $\alpha _{j}\in \mathbb {A} _{K}$ are unique. The map $\alpha \mapsto \alpha _{j}$ is continuous. We define $\alpha _{ij},$ depending on $\alpha ,$ via the equations:

${\begin{aligned}\alpha \omega _{1}&=\sum _{j=1}^{n}\alpha _{1j}\omega _{j}\\&\vdots \\\alpha \omega _{n}&=\sum _{j=1}^{n}\alpha _{nj}\omega _{j}\end{aligned}}$

Now, we define the trace and norm of $\alpha$ as:

${\begin{aligned}\operatorname {Tr} _{L/K}(\alpha )&:=\operatorname {Tr} ((\alpha _{ij})_{i,j})=\sum _{i=1}^{n}\alpha _{ii}\\N_{L/K}(\alpha )&:=N((\alpha _{ij})_{i,j})=\det((\alpha _{ij})_{i,j})\end{aligned}}$

These are the trace and the determinant of the linear map

${\begin{cases}\mathbb {A} _{L}\to \mathbb {A} _{L}\\x\mapsto \alpha x\end{cases}}$

They are continuous maps on the adele ring and they fulfil the usual equations:

${\begin{aligned}\operatorname {Tr} _{L/K}(\alpha +\beta )&=\operatorname {Tr} _{L/K}(\alpha )+\operatorname {Tr} _{L/K}(\beta )&&\forall \alpha ,\beta \in \mathbb {A} _{L}\\\operatorname {Tr} _{L/K}(\operatorname {con} (\alpha ))&=n\alpha &&\forall \alpha \in \mathbb {A} _{K}\\N_{L/K}(\alpha \beta )&=N_{L/K}(\alpha )N_{L/K}(\beta )&&\forall \alpha ,\beta \in \mathbb {A} _{L}\\N_{L/K}(\operatorname {con} (\alpha ))&=\alpha ^{n}&&\forall \alpha \in \mathbb {A} _{K}\end{aligned}}$

Furthermore, for $\alpha \in L,$$\operatorname {Tr} _{L/K}(\alpha )$ and $N_{L/K}(\alpha )$ are identical to the trace and norm of the field extension $L/K.$ For a tower of fields $M/L/K,$ we have:

${\begin{aligned}\operatorname {Tr} _{L/K}(\operatorname {Tr} _{M/L}(\alpha ))&=\operatorname {Tr} _{M/K}(\alpha )&&\forall \alpha \in \mathbb {A} _{M}\\N_{L/K}(N_{M/L}(\alpha ))&=N_{M/K}(\alpha )&&\forall \alpha \in \mathbb {A} _{M}\end{aligned}}$

Moreover, it can be proven that:

${\begin{aligned}\operatorname {Tr} _{L/K}(\alpha )&=\left(\sum _{w|v}\operatorname {Tr} _{L_{w}/K_{v}}(\alpha _{w})\right)_{v}&&\forall \alpha \in \mathbb {A} _{L}\\N_{L/K}(\alpha )&=\left(\prod _{w|v}N_{L_{w}/K_{v}}(\alpha _{w})\right)_{v}&&\forall \alpha \in \mathbb {A} _{L}\end{aligned}}$

Properties of the adele ring

Theorem. For every set of places $S,\mathbb {A} _{K,S}$ is a locally compact topological ring.

Remark. The result above also holds for the adele ring of vector-spaces and algebras over $K.$

Theorem. $K$ is discrete and cocompact in $\mathbb {A} _{K}.$ In particular, $K$ is closed in $\mathbb {A} _{K}.$

Proof. We prove the case $K=\mathbb {Q} .$ To show $\mathbb {Q} \subset \mathbb {A} _{\mathbb {Q} }$ is discrete it is sufficient to show the existence of a neighbourhood of $0$ which contains no other rational number. The general case follows via translation. Define

$U:=\left\{(\alpha _{p})_{p}\left|\forall p<\infty :|\alpha _{p}|_{p}\leq 1\quad {\text{and}}\quad |\alpha _{\infty }|_{\infty }<1\right.\right\}={\widehat {\mathbb {Z} }}\times (-1,1).$

$U$ is an open neighbourhood of $0\in \mathbb {A} _{\mathbb {Q} }.$ We claim $U\cap \mathbb {Q} =\{0\}.$ Let $\beta \in U\cap \mathbb {Q} ,$ then $\beta \in \mathbb {Q}$ and $|\beta |_{p}\leq 1$ for all $p$ and therefore $\beta \in \mathbb {Z} .$ Additionally, we have $\beta \in (-1,1)$ and therefore $\beta =0.$ Next, we show compactness, define:

$W:=\left\{(\alpha _{p})_{p}\left|\forall p<\infty :|\alpha _{p}|_{p}\leq 1\quad {\text{and}}\quad |\alpha _{\infty }|_{\infty }\leq {\frac {1}{2}}\right.\right\}={\widehat {\mathbb {Z} }}\times \left[-{\frac {1}{2}},{\frac {1}{2}}\right].$

We show each element in $\mathbb {A} _{\mathbb {Q} }/\mathbb {Q}$ has a representative in $W,$ that is for each $\alpha \in \mathbb {A} _{\mathbb {Q} },$ there exists $\beta \in \mathbb {Q}$ such that $\alpha -\beta \in W.$ Let $\alpha =(\alpha _{p})_{p}\in \mathbb {A} _{\mathbb {Q} },$ be arbitrary and $p$ be a prime for which $|\alpha _{p}|>1.$ Then there exists $r_{p}=z_{p}/p^{x_{p}}$ with $z_{p}\in \mathbb {Z} ,x_{p}\in \mathbb {N}$ and $|\alpha _{p}-r_{p}|\leq 1.$ Replace $\alpha$ with $\alpha -r_{p}$ and let $q\neq p$ be another prime. Then:

$\left|\alpha _{q}-r_{p}\right|_{q}\leq \max \left\{|a_{q}|_{q},|r_{p}|_{q}\right\}\leq \max \left\{|a_{q}|_{q},1\right\}\leq 1.$

Next we claim:

$|\alpha _{q}-r_{p}|_{q}\leq 1\Longleftrightarrow |\alpha _{q}|_{q}\leq 1.$

The reverse implication is trivially true. The implication is true, because the two terms of the strong triangle inequality are equal if the absolute values of both integers are different. As a consequence, the (finite) set of primes for which the components of $\alpha$ are not in $\mathbb {Z} _{p}$ is reduced by 1. With iteration, we deduce there exists $r\in \mathbb {Q}$ such that $\alpha -r\in {\widehat {\mathbb {Z} }}\times \mathbb {R} .$ Now we select $s\in \mathbb {Z}$ such that $\alpha _{\infty }-r-s\in \left[-{\tfrac {1}{2}},{\tfrac {1}{2}}\right].$ Then $\alpha -(r+s)\in W.$ The continuous projection $\pi :W\to \mathbb {A} _{\mathbb {Q} }/\mathbb {Q}$ is surjective, therefore $\mathbb {A} _{\mathbb {Q} }/\mathbb {Q} ,$ as the continuous image of a compact set, is compact.

Corollary. Let $E$ be a finite-dimensional vector-space over $K.$ Then $E$ is discrete and cocompact in $\mathbb {A} _{E}.$

Theorem. We have the following:

• $\mathbb {A} _{\mathbb {Q} }=\mathbb {Q} +\mathbb {A} _{\mathbb {Z} }.$
• $\mathbb {Z} =\mathbb {Q} \cap \mathbb {A} _{\mathbb {Z} }.$
• $\mathbb {A} _{\mathbb {Q} }/\mathbb {Z}$ is a divisible group.
• $\mathbb {Q} \subset \mathbb {A} _{\mathbb {Q} ,{\text{fin}}}$ is dense.

Proof. The first two equations can be proved in an elementary way.

By definition $\mathbb {A} _{\mathbb {Q} }/\mathbb {Z}$ is divisible if for any $n\in \mathbb {N}$ and $y\in \mathbb {A} _{\mathbb {Q} }/\mathbb {Z}$ the equation $nx=y$ has a solution $x\in \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} .$ It is sufficient to show $\mathbb {A} _{\mathbb {Q} }$ is divisible but this is true since $\mathbb {A} _{\mathbb {Q} }$ is a field with positive characteristic in each coordinate.

For the last statement note that $\mathbb {A} _{\mathbb {Q} ,{\text{fin}}}=\mathbb {Q} {\widehat {\mathbb {Z} }},$ as we can reach the finite number of denominators in the coordinates of the elements of $\mathbb {A} _{\mathbb {Q} ,{\text{fin}}}$ through an element $q\in \mathbb {Q} .$ As a consequence, it is sufficient to show $\mathbb {Z} \subset {\widehat {\mathbb {Z} }}$ is dense, that is each open subset $V\subset {\widehat {\mathbb {Z} }}$ contains an element of $\mathbb {Z} .$ Without loss of generality, we can assume

$V=\prod _{p\in E}\left(a_{p}+p^{l_{p}}\mathbb {Z} _{p}\right)\times \prod _{p\notin E}\mathbb {Z} _{p},$

because $(p^{m}\mathbb {Z} _{p})_{m\in \mathbb {N} }$ is a neighbourhood system of $0$ in $\mathbb {Z} _{p}.$ By Chinese Remainder Theorem there exists $l\in \mathbb {Z}$ such that $l\equiv a_{p}{\bmod {p}}^{l_{p}}.$ Since powers of distinct primes are coprime, $l\in V$ follows.

Remark. $\mathbb {A} _{\mathbb {Q} }/\mathbb {Z}$ is not uniquely divisible. Let $y=(0,0,\ldots )+\mathbb {Z} \in \mathbb {A} _{\mathbb {Q} }/\mathbb {Z}$ and $n\geq 2$ be given. Then

${\begin{aligned}x_{1}&=(0,0,\ldots )+\mathbb {Z} \\x_{2}&=\left({\tfrac {1}{n}},{\tfrac {1}{n}},\ldots \right)+\mathbb {Z} \end{aligned}}$

both satisfy the equation $nx=y$ and clearly $x_{1}\neq x_{2}$ ($x_{2}$ is well-defined, because only finitely many primes divide $n$). In this case, being uniquely divisible is equivalent to being torsion-free, which is not true for $\mathbb {A} _{\mathbb {Q} }/\mathbb {Z}$ since $nx_{2}=0,$ but $x_{2}\neq 0$ and $n\neq 0.$

Remark. The fourth statement is a special case of the strong approximation theorem.

Haar measure on the adele ring

Definition. A function $f:\mathbb {A} _{K}\to \mathbb {C}$ is called simple if $\textstyle f=\prod _{v}f_{v},$ where $f_{v}:K_{v}\to \mathbb {C}$ are measurable and $f_{v}=\mathbf {1} _{O_{v}}$ for almost all $v.$

Theorem. Since $\mathbb {A} _{K}$ is a locally compact group with addition, there is an additive Haar measure $dx$ on $\mathbb {A} _{K}.$ This measure can be normalized such that every integrable simple function $\textstyle f=\prod _{v}f_{v}$ satisfies:

$\int _{\mathbb {A} _{K}}f\,dx=\prod _{v}\int _{K_{v}}f_{v}\,dx_{v},$

where for $v<\infty ,dx_{v}$ is the measure on $K_{v}$ such that $O_{v}$ has unit measure and $dx_{\infty }$ is the Lebesgue measure. The product is finite, i.e. almost all factors are equal to one.

The idele group

Definition. We define the idele group of $K$ as the group of units of the adele ring of $K,$ that is $I_{K}:=\mathbb {A} _{K}^{\times }.$ The elements of the idele group are called the ideles of $K.$

Remark. We would like to equip $I_{K}$ with a topology so that it becomes a topological group. The subset topology inherited from $\mathbb {A} _{K}$ is not a suitable candidate since the group of units of a topological ring equipped with subset topology may not be a topological group. For example the inverse map in $\mathbb {A} _{\mathbb {Q} }$ is not continuous. The sequence

${\begin{aligned}x_{1}&=(2,1,\ldots )\\x_{2}&=(1,3,1,\ldots )\\x_{3}&=(1,1,5,1,\ldots )\\&\vdots \end{aligned}}$

converges to $1\in \mathbb {A} _{\mathbb {Q} }.$ To see this let $U$ be neighbourhood of $0,$ without loss of generality we can assume:

$U=\prod _{p\leq N}U_{p}\times \prod _{p>N}\mathbb {Z} _{p}$

Since $(x_{n})_{p}-1\in \mathbb {Z} _{p}$ for all $p,$ $x_{n}-1\in U$ for $n$ large enough. However as we saw above the inverse of this sequence does not converge in $\mathbb {A} _{\mathbb {Q} }.$

Lemma. Let $R$ be a topological ring. Define:

${\begin{cases}\iota :R^{\times }\to R\times R\\x\mapsto (x,x^{-1})\end{cases}}$

Equipped with the topology induced from the product on topology on $R\times R$ and $\iota ,R^{\times }$ is a topological group and the inclusion map $R^{\times }\subset R$ is continuous. It is the coarsest topology, emerging from the topology on $R,$ that makes $R^{\times }$ a topological group.

Proof. Since $R$ is a topological ring, it is sufficient to show that the inverse map is continuous. Let $U\subset R^{\times }$ be open, then $U\times U^{-1}\subset R\times R$ is open. We have to show $U^{-1}\subset R^{\times }$ is open or equivalently, that $U^{-1}\times (U^{-1})^{-1}=U^{-1}\times U\subset R\times R$ is open. But this is the condition above.

We equip the idele group with the topology defined in the Lemma making it a topological group.

Definition. For $S$ a subset of places of $K$ set: $I_{K,S}:=\mathbb {A} _{K,S}^{\times },I_{K}^{S}:=(\mathbb {A} _{K}^{S})^{\times }.$

Lemma. The following identities of topological groups hold:

${\begin{aligned}I_{K,S}&={\prod _{v\in S}}^{'}K_{v}^{\times }\\I_{K}^{S}&={\prod _{v\notin S}}^{'}K_{v}^{\times }\\I_{K}&={\prod _{v}}^{'}K_{v}^{\times }\end{aligned}}$

where the restricted product has the restricted product topology, which is generated by restricted open rectangles of the form

$\prod _{v\in E}U_{v}\times \prod _{v\notin E}O_{v}^{\times },$

where $E$ is a finite subset of the set of all places and $U_{v}\subset K_{v}^{\times }$ are open sets.

Proof. We prove the identity for $I_{K},$ the other two follow similarly. First we show the two sets are equal:

${\begin{aligned}I_{K}&=\{x=(x_{v})_{v}\in \mathbb {A} _{K}:\exists y=(y_{v})_{v}\in \mathbb {A} _{K}:xy=1\}\\&=\{x=(x_{v})_{v}\in \mathbb {A} _{K}:\exists y=(y_{v})_{v}\in \mathbb {A} _{K}:x_{v}\cdot y_{v}=1\quad \forall v\}\\&=\{x=(x_{v})_{v}:x_{v}\in K_{v}^{\times }\forall v{\text{ and }}x_{v}\in O_{v}^{\times }{\text{ for almost all }}v\}\\&={\prod _{v}}^{'}K_{v}^{\times }\end{aligned}}$

In going from line 2 to 3, $x$ as well as $x^{-1}=y$ have to be in $\mathbb {A} _{K},$ meaning $x_{v}\in O_{v}$ for almost all $v$ and $x_{v}^{-1}\in O_{v}$ for almost all $v.$ Therefore, $x_{v}\in O_{v}^{\times }$ for almost all $v.$

Now, we can show the topology on the left hand side equals the topology on the right hand side. Obviously, every open restricted rectangle is open in the topology of the idele group. On the other hand, for a given $U\subset I_{K},$ which is open in the topology of the idele group, meaning $U\times U^{-1}\subset \mathbb {A} _{K}\times \mathbb {A} _{K}$ is open, so for each $u\in U$ there exists an open restricted rectangle, which is a subset of $U$ and contains $u.$ Therefore, $U$ is the union of all these restricted open rectangles and therefore is open in the restricted product topology.

Lemma. For each set of places, $S,I_{K,S}$ is a locally compact topological group.

Proof. The local compactness follows from the description of $I_{K,S}$ as a restricted product. It being a topological group follows from the above discussion on the group of units of a topological ring.

A neighbourhood system of $1\in \mathbb {A} _{K}(P_{\infty })^{\times }$ is a neighbourhood system of $1\in I_{K}.$ Alternatively, we can take all sets of the form:

$\prod _{v}U_{v},$

where $U_{v}$ is a neighbourhood of $1\in K_{v}^{\times }$ and $U_{v}=O_{v}^{\times }$ for almost all $v.$

Since the idele group is a locally compact, there exists a Haar measure $d^{\times }x$ on it. This can be normalised, so that

$\int _{I_{K,{\text{fin}}}}\mathbf {1} _{\widehat {O}}\,d^{\times }x=1.$

This is the normalisation used for the finite places. In this equations, $I_{K,{\text{fin}}}$ is the finite idele group, meaning the group of units of the finite adele ring. For the infinite places, we use the multiplicative lebesgue measure ${\tfrac {dx}{|x|}}.$

The idele group of a finite extension

Lemma. Let $L/K$ be a finite extension. Then:

$I_{L}={\prod _{v}}^{'}L_{v}^{\times }.$

where the restricted product is with respect to ${\widetilde {O_{v}}}^{\times }.$

Lemma. There is a canonical embedding of $I_{K}$ in $I_{L}.$

Proof. We map $a=(a_{v})_{v}\in I_{K}$ to $a'=(a'_{w})_{w}\in I_{L}$ with the property $a'_{w}=a_{v}\in K_{v}^{\times }\subset L_{w}^{\times }$ for $w|v.$ Therefore, $I_{K}$ can be seen as a subgroup of $I_{L}.$ An element $a=(a_{w})_{w}\in I_{L}$ is in this subgroup if and only if his components satisfy the following properties: $a_{w}\in K_{v}^{\times }$ for $w|v$ and $a_{w}=a_{w'}$ for $w|v$ and $w'|v$ for the same place $v$ of $K.$

The case of vector-spaces and algebras

The idele group of an algebra

Let $A$ be a finite-dimensional algebra over $K.$ Since $\mathbb {A} _{A}^{\times }$ is not a topological group with the subset-topology in general, we equip $\mathbb {A} _{A}^{\times }$ with the topology similar to $I_{K}$ above and call $\mathbb {A} _{A}^{\times }$ the idele group. The elements of the idele group are called idele of $A.$

Proposition. Let $\alpha$ be a finite subset of $A,$ containing a basis of $A$ over $K.$ For each finite place $v$ of $K,$ let $\alpha _{v}$ be the $O_{v}$-module generated by $\alpha$ in $A_{v}.$ There exists a finite set of places $P_{0}$ containing $P_{\infty }$ such that for all $v\notin P_{0},$$\alpha _{v}$ is a compact subring of $A_{v}.$ Furthermore, $\alpha _{v}$ contains $A_{v}^{\times }.$ For each $v,A_{v}^{\times }$ is an open subset of $A_{v}$ and the map $x\mapsto x^{-1}$ is continuous on $A_{v}^{\times }.$ As a consequence $x\mapsto (x,x^{-1})$ maps $A_{v}^{\times }$ homeomorphically on its image in $A_{v}\times A_{v}.$ For each $v\notin P_{0},$ the $\alpha _{v}^{\times }$ are the elements of $A_{v}^{\times },$ mapping in $\alpha _{v}\times \alpha _{v}$ with the function above. Therefore, $\alpha _{v}^{\times }$ is an open and compact subgroup of $A_{v}^{\times }.$

Alternative characterisation of the idele group

Proposition. Let $P\supset P_{\infty }$ be a finite set of places. Then

$\mathbb {A} _{A}(P,\alpha )^{\times }:=\prod _{v\in P}A_{v}^{\times }\times \prod _{v\notin P}\alpha _{v}^{\times }$

is an open subgroup of $\mathbb {A} _{A}^{\times },$ where $\mathbb {A} _{A}^{\times }$ is the union of all $\mathbb {A} _{A}(P,\alpha )^{\times }.$

Corollary. In the special case of $A=K$ for each finite set of places $P\supset P_{\infty },$

$\mathbb {A} _{K}(P)^{\times }=\prod _{v\in P}K_{v}^{\times }\times \prod _{v\notin P}O_{v}^{\times }$

is an open subgroup of $\mathbb {A} _{K}^{\times }=I_{K}.$ Furthermore, $I_{K}$ is the union of all $\mathbb {A} _{K}(P)^{\times }.$

Norm on the idele group

We want to transfer the trace and the norm from the adele ring to the idele group. It turns out the trace can't be transferred so easily. However, it is possible to transfer the norm from the adele ring to the idele group. Let $\alpha \in I_{K}.$ Then $\operatorname {con} _{L/K}(\alpha )\in I_{L}$ and therefore, we have in injective group homomorphism

$\operatorname {con} _{L/K}:I_{K}\hookrightarrow I_{L}.$

Since $\alpha \in I_{L},$ it is invertible, $N_{L/K}(\alpha )$ is invertible too, because $(N_{L/K}(\alpha ))^{-1}=N_{L/K}(\alpha ^{-1}).$ Therefore $N_{L/K}(\alpha )\in I_{K}.$ As a consequence, the restriction of the norm-function introduces a continuous function:

$N_{L/K}:I_{L}\to I_{K}.$

The Idele class group

Lemma. There is natural embedding of $K^{\times }$ into $I_{K,S}$ given by the diagonal map: $a\mapsto (a,a,a,\ldots ).$

Proof. Since $K^{\times }$ is a subset of $K_{v}^{\times }$ for all $v,$ the embedding is well-defined and injective.

Corollary. $A^{\times }$ is a discrete subgroup of $\mathbb {A} _{A}^{\times }.$

Defenition. In analogy to the ideal class group, the elements of $K^{\times }$ in $I_{K}$ are called principal ideles of $I_{K}.$ The quotient group $C_{K}:=I_{K}/K^{\times }$ is called idele class group of $K.$ This group is related to the ideal class group and is a central object in class field theory.

Remark. $K^{\times }$ is closed in $I_{K},$ therefore $C_{K}$ is a locally compact topological group and a Hausdorff space.

Lemma. Let $L/K$ be a finite extension. The embedding $I_{K}\to I_{L}$ induces an injective map:

${\begin{cases}C_{K}\to C_{L}\\\alpha K^{\times }\mapsto \alpha L^{\times }\end{cases}}$

Properties of the idele group

Absolute value on $I_{K}$ and $1$-idele

Definition. For $\alpha =(\alpha _{v})_{v}\in I_{K}$ define: $\textstyle |\alpha |:=\prod _{v}|\alpha _{v}|_{v}.$ Since $\alpha$ is an idele this product is finite and therefore well-defined.

Remark. The definition can be extended to $\mathbb {A} _{K}$ by allowing infinite products. However these infinite products vanish and so $|\cdot |$ vanishes on $\mathbb {A} _{K}\setminus I_{K}.$ We will use $|\cdot |$ to denote both the function on $I_{K}$ and $\mathbb {A} _{K}.$

Theorem. $|\cdot |:I_{K}\to \mathbb {R} _{+}$ is a continuous group homomorphism.

Proof. Let $\alpha ,\beta \in I_{K}.$

${\begin{aligned}|\alpha \cdot \beta |&=\prod _{v}|(\alpha \cdot \beta )_{v}|_{v}\\&=\prod _{v}|\alpha _{v}\cdot \beta _{v}|_{v}\\&=\prod _{v}(|\alpha _{v}|_{v}\cdot |\beta _{v}|_{v})\\&=\left(\prod _{v}|\alpha _{v}|_{v}\right)\cdot \left(\prod _{v}|\beta _{v}|_{v}\right)\\&=|\alpha |\cdot |\beta |\end{aligned}}$

where we use that all products are finite. The map is continuous which can be seen using an argument dealing with sequences. This reduces the problem to whether $|\cdot |$ is continuous on $K_{v}.$ However, this is clear, because of the reverse triangle inequality.

Definition. We define the set of $1$-idele as:

$I_{K}^{1}:=\{x\in I_{K}:|x|=1\}=\ker(|\cdot |).$

$I_{K}^{1}$ is a subgroup of $I_{K}.$ Since $I_{K}^{1}=|\cdot |^{-1}(\{1\}),$ it is a closed subset of $\mathbb {A} _{K}.$ Finally the $\mathbb {A} _{K}$-topology on $I_{K}^{1}$ equals the subset-topology of $I_{K}$ on $I_{K}^{1}.$

Artin's Product Formula. $|k|=1$ for all $k\in K^{\times }.$

Proof. We prove the formula for number fields, the case of global function fields can be proved similarly. Let $K$ be a number field and $a\in K^{\times }.$ We have to show:

$\prod _{v}|a|_{v}=1.$

For a finite place $v$ for which the corresponding prime ideal ${\mathfrak {p}}_{v}$ does not divide $(a)$ we have $v(a)=0$ and therefore $|a|_{v}=1.$ This is valid for almost all ${\mathfrak {p}}_{v}.$ We have:

${\begin{aligned}\prod _{v}|a|_{v}&=\prod _{p\leq \infty }\prod _{v|p}|a|_{v}\\&=\prod _{p\leq \infty }\prod _{v|p}|N_{K_{v}/\mathbb {Q} _{p}}(a)|_{p}\\&=\prod _{p\leq \infty }|N_{K/\mathbb {Q} }(a)|_{p}\end{aligned}}$

In going from line 1 to line 2, we used the identity $|a|_{w}=|N_{L_{w}/K_{v}}(a)|_{v},$ where $v$ is a place of $K$ and $w$ is a place of $L,$ lying above $v.$ Going from line 2 to line 3, we use a property of the norm. We note the norm is in $\mathbb {Q}$ so without loss of generality we can assume $a\in \mathbb {Q} .$ Then $a$ possesses a unique integer factorisation:

$a=\pm \prod _{p<\infty }p^{v_{p}},$

where $v_{p}\in \mathbb {Z}$ is $0$ for almost all $p.$ By Ostrowski's theorem all absolute values on $\mathbb {Q}$ are equivalent to the real absolute value $|\cdot |_{\infty }$ or a $p$-adic absolute value. Therefore:

${\begin{aligned}|a|&=\left(\prod _{p<\infty }|a|_{p}\right)\cdot |a|_{\infty }\\&=\left(\prod _{p<\infty }p^{-v_{p}}\right)\cdot \left(\prod _{p<\infty }p^{v_{p}}\right)\\&=1\end{aligned}}$

Lemma. There exists a constant $C,$ depending only on $K,$ such that for every $\alpha =(\alpha _{v})_{v}\in \mathbb {A} _{K}$ satisfying $\textstyle \prod _{v}|\alpha _{v}|_{v}>C,$ there exists $\beta \in K^{\times }$ such that $|\beta _{v}|_{v}\leq |\alpha _{v}|_{v}$ for all $v.$

Corollary. Let $v_{0}$ be a place of $K$ and let $\delta _{v}>0$ be given for all $v\neq v_{0}$ with the property $\delta _{v}=1$ for almost all $v.$ Then there exists $\beta \in K^{\times },$ so that $|\beta |\leq \delta _{v}$ for all $v\neq v_{0}.$

Proof. Let $C$ be the constant from the lemma. Let $\pi _{v}$ be a uniformizing element of $O_{v}.$ Define the adele $\alpha =(\alpha _{v})_{v}$ via $\alpha _{v}:=\pi _{v}^{k_{v}}$ with $k_{v}\in \mathbb {Z}$ minimal, so that $|\alpha _{v}|_{v}\leq \delta _{v}$ for all $v\neq v_{0}.$ Then $k_{v}=0$ for almost all $v.$ Define $\alpha _{v_{0}}:=\pi _{v_{0}}^{k_{v_{0}}}$ with $k_{v_{0}}\in \mathbb {Z} ,$ so that $\textstyle \prod _{v}|\alpha _{v}|_{v}>C.$ This works, because $k_{v}=0$ for almost all $v.$ By the Lemma there exists $\beta \in K^{\times },$ so that $|\beta |_{v}\leq |\alpha _{v}|_{v}\leq \delta _{v}$ for all $v\neq v_{0}.$

Theorem. $K^{\times }$ is discrete and cocompact in $I_{K}^{1}.$

Proof. Since $K^{\times }$ is discrete in $I_{K}$ it is also discrete in $I_{K}^{1}.$ To prove the compactness of $I_{K}^{1}/K^{\times }$ let $C$ is the constant of the Lemma and suppose $\alpha \in \mathbb {A} _{K}$ satisfying $\textstyle \prod _{v}|\alpha _{v}|_{v}>C$ is given. Define:

$W_{\alpha }:=\left\{\xi =(\xi _{v})_{v}\in \mathbb {A} _{K}||\xi _{v}|_{v}\leq |\alpha _{v}|_{v}{\text{ for all }}v\right\}.$

Clearly $W_{\alpha }$ is compact. We claim the natural projection $W_{\alpha }\cap I_{K}^{1}\to I_{K}^{1}/K^{\times }$ is surjective. Let $\beta =(\beta _{v})_{v}\in I_{K}^{1}$ be arbitrary, then:

$|\beta |=\prod _{v}|\beta _{v}|_{v}=1,$

and therefore

$\prod _{v}|\beta _{v}^{-1}|_{v}=1.$

It follows that

$\prod _{v}|\beta _{v}^{-1}\alpha _{v}|_{v}=\prod _{v}|\alpha _{v}|_{v}>C.$

By the Lemma there exists $\eta \in K^{\times }$ such that $|\eta |_{v}\leq |\beta _{v}^{-1}\alpha _{v}|_{v}$ for all $v,$ and therefore $\eta \beta \in W_{\alpha }$ proving the surjectivity of the natural projection. Since it is also continuous the compactness follows.

Theorem. There is a canonical isomorphism $I_{\mathbb {Q} }^{1}/\mathbb {Q} ^{\times }\cong {\widehat {\mathbb {Z} }}^{\times }.$ Furthermore, ${\widehat {\mathbb {Z} }}^{\times }\times \{1\}\subset I_{\mathbb {Q} }^{1}$ is a set of representatives for $I_{\mathbb {Q} }^{1}/\mathbb {Q} ^{\times }$ and ${\widehat {\mathbb {Z} }}^{\times }\times (0,\infty )\subset I_{\mathbb {Q} }$ is a set of representatives for $I_{\mathbb {Q} }/\mathbb {Q} ^{\times }.$

Proof. Consider the map

${\begin{cases}\phi :{\widehat {\mathbb {Z} }}^{\times }\to I_{\mathbb {Q} }^{1}/\mathbb {Q} ^{\times }\\(a_{p})_{p}\mapsto ((a_{p})_{p},1)\mathbb {Q} ^{\times }\end{cases}}$

This map is well-defined, since $|a_{p}|_{p}=1$ for all $p$ and therefore $\textstyle \left(\prod _{p<\infty }|a_{p}|_{p}\right)\cdot 1=1.$ Obviously $\phi$ is a continuous group homomorphism. Now suppose $((a_{p})_{p},1)\mathbb {Q} ^{\times }=((b_{p})_{p},1)\mathbb {Q} ^{\times }.$ Then there exists $q\in \mathbb {Q} ^{\times }$ such that $((a_{p})_{p},1)q=((b_{p})_{p},1).$ By considering the infinite place we see $q=1$ proving injectivity. To show surjectivity let $((\beta _{p})_{p},\beta _{\infty })\mathbb {Q} ^{\times }\in I_{\mathbb {Q} }^{1}/\mathbb {Q} ^{\times }.$ The absolute value of this element is $1$ and therefore

$|\beta _{\infty }|_{\infty }={\frac {1}{\prod _{p}|\beta _{p}|_{p}}}\in \mathbb {Q} .$

Hence $\beta _{\infty }\in \mathbb {Q}$ and we have:

$((\beta _{p})_{p},\beta _{\infty })\mathbb {Q} ^{\times }=\left(\left({\frac {\beta _{p}}{\beta _{\infty }}}\right)_{p},1\right)\mathbb {Q} ^{\times }.$

Since

$\forall p:\qquad \left|{\frac {\beta _{p}}{\beta _{\infty }}}\right|_{p}=1,$

we conclude $\phi$ is surjective.

Theorem. The absolute value function induces the following isomorphisms of topological groups:

${\begin{aligned}I_{\mathbb {Q} }&\cong I_{\mathbb {Q} }^{1}\times (0,\infty )\\I_{\mathbb {Q} }^{1}&\cong I_{\mathbb {Q} ,{\text{fin}}}\times \{\pm 1\}.\end{aligned}}$

Proof. The isomorphisms are given by:

${\begin{cases}\psi :I_{\mathbb {Q} }\to I_{\mathbb {Q} }^{1}\times (0,\infty )\\a=(a_{\text{fin}},a_{\infty })\mapsto \left(a_{\text{fin}},{\frac {a_{\infty }}{|a|}},|a|\right)\end{cases}}\qquad {\text{and}}\qquad {\begin{cases}{\widetilde {\psi }}:I_{\mathbb {Q} ,{\text{fin}}}\times \{\pm 1\}\to I_{\mathbb {Q} }^{1}\\(a_{\text{fin}},\varepsilon )\mapsto \left(a_{\text{fin}},{\frac {\varepsilon }{|a_{\text{fin}}|}}\right)\end{cases}}$

Relation between ideal class group and idele class group

Theorem. Let $K$ be a number field with ring of integers $O,$ group of fractional ideals $J_{K},$ and ideal class group $\operatorname {Cl} _{K}=J_{K}/K^{\times }.$ We have the following isomorphisms

${\begin{aligned}J_{K}&\cong I_{K,{\text{fin}}}/{\widehat {O}}^{\times }\\\operatorname {Cl} _{K}&\cong C_{K,{\text{fin}}}/{\widehat {O}}^{\times }K^{\times }\\\operatorname {Cl} _{K}&\cong C_{K}/\left({\widehat {O}}^{\times }\times \prod _{v|\infty }K_{v}^{\times }\right)K^{\times }\end{aligned}}$

where we have defined $C_{K,{\text{fin}}}:=I_{K,{\text{fin}}}/K^{\times }.$

Proof. Let $v$ be a finite place of $K$ and let $|\cdot |_{v}$ be a representative of the equivalence class $v.$ Define

${\mathfrak {p}}_{v}:=\{x\in O:|x|_{v}<1\}.$

Then ${\mathfrak {p}}_{v}$ is a prime ideal in $O.$ The map $v\mapsto {\mathfrak {p}}_{v}$ is a bijection between finite places of $K$ and non-zero prime ideals of $O.$ The inverse is given as follows: a prime ideal ${\mathfrak {p}}$ is mapped to the valuation $v_{\mathfrak {p}},$ given by

${\begin{aligned}v_{\mathfrak {p}}(x)&:=\max\{k\in \mathbb {N} _{0}:x\in {\mathfrak {p}}^{k}\}\quad \forall x\in O^{\times }\\v_{\mathfrak {p}}\left({\frac {x}{y}}\right)&:=v_{\mathfrak {p}}(x)-v_{\mathfrak {p}}(y)\quad \forall x,y\in O^{\times }\end{aligned}}$

The following map is well-defined:

${\begin{cases}(\cdot ):I_{K,{\text{fin}}}\to J_{K}\\\alpha =(\alpha _{v})_{v}\mapsto \prod _{v<\infty }{\mathfrak {p}}_{v}^{v(\alpha _{v})},\end{cases}}$

The map $(\cdot )$ is obviously a surjective homomorphism and $\ker((\cdot ))={\widehat {O}}^{\times }.$ The first isomorphism follows from fundamental theorem on homomorphism. Now, we divide both sides by $K^{\times }.$ This is possible, because

${\begin{aligned}(\alpha )&=((\alpha ,\alpha ,\dotsc ))\\&=\prod _{v<\infty }{\mathfrak {p}}_{v}^{v(\alpha )}\\&=(\alpha )&&{\text{ for all }}\alpha \in K^{\times }.\end{aligned}}$

Please, note the abuse of notation: On the left hand side in line 1 of this chain of equations, $(\cdot )$ stands for the map defined above. Later, we use the embedding of $K^{\times }$ into $I_{K,{\text{fin}}}.$ In line 2, we use the definition of the map. Finally, we use that $O$ is a Dedekind domain and therefore each ideal can be written as a product of prime ideals. In other words, the map $(\cdot )$ is a $K^{\times }$-equivariant group homomorphism. As a consequence, the map above induces a surjective homomorphism

${\begin{cases}\phi :C_{K,{\text{fin}}}\to \operatorname {Cl} _{K}\\\alpha K^{\times }\mapsto (\alpha )K^{\times }\end{cases}}$

To prove the second isomorphism we have to show $\ker(\phi )={\widehat {O}}^{\times }K^{\times }.$ Consider $\xi =(\xi _{v})_{v}\in {\widehat {O}}^{\times }.$ Then $\textstyle \phi (\xi K^{\times })=\prod _{v}{\mathfrak {p}}_{v}^{v(\xi _{v})}K^{\times }=K^{\times },$ because $v(\xi _{v})=0$ for all $v.$ On the other hand, consider $\xi K^{\times }\in C_{K,{\text{fin}}}$ with $\phi (\xi K^{\times })=OK^{\times },$ which allows to write $\textstyle \prod _{v}{\mathfrak {p}}_{v}^{v(\xi _{v})}K^{\times }=OK^{\times }.$ As a consequence, there exists a representative, such that: $\textstyle \prod _{v}{\mathfrak {p}}_{v}^{v(\xi '_{v})}=O.$ Consequently, $\xi '\in {\widehat {O}}^{\times }$ and therefore $\xi K^{\times }=\xi 'K^{\times }\in {\widehat {O}}^{\times }K^{\times }.$ We have proved the second isomorphism of the theorem.

For the last isomorphism note that $\phi$ induces a surjective group homomorphism ${\widetilde {\phi }}:C_{K}\to \operatorname {Cl} _{K}$ with

$\ker({\widetilde {\phi }})=\left({\widehat {O}}^{\times }\times \prod _{v|\infty }K_{v}^{\times }\right)K^{\times }.$

Remark. Consider $I_{K,{\text{fin}}}$ with the idele topology and equip $J_{K},$ with the discrete topology. Since $(\{{\mathfrak {a}}\})^{-1}$ is open for each ${\mathfrak {a}}\in J_{K},(\cdot )$ is continuous. It stands, that $(\{{\mathfrak {a}}\})^{-1}=\alpha {\widehat {O}}^{\times }$ is open, where $\alpha =(\alpha _{v})_{v}\in \mathbb {A} _{K,{\text{fin}}},$ so that $\textstyle {\mathfrak {a}}=\prod _{v}{\mathfrak {p}}_{v}^{v(\alpha _{v})}.$

Decomposition of $I_{K}$ and $C_{K}$

Theorem.

${\begin{aligned}I_{K}&\cong I_{K}^{1}\times M:\quad {\begin{cases}M\subset I_{K}{\text{ discrete and }}M\cong \mathbb {Z} &\operatorname {char} (K)>0\\M\subset I_{K}{\text{ closed and }}M\cong \mathbb {R} _{+}&\operatorname {char} (K)=0\end{cases}}\\C_{K}&\cong I_{K}^{1}/K^{\times }\times N:\quad {\begin{cases}N=\mathbb {Z} &\operatorname {char} (K)>0\\N=\mathbb {R} _{+}&\operatorname {char} (K)=0\end{cases}}\end{aligned}}$

Proof. $\operatorname {char} (K)=p>0.$ For each place $v$ of $K,\operatorname {char} (K_{v})=p,$ so that for all $x\in K_{v}^{\times },$ $|x|_{v}$ belongs to the subgroup of $\mathbb {R} _{+},$ generated by $p.$ Therefore for each $z\in I_{K},$ $|z|$ is in the subgroup of $\mathbb {R} _{+},$ generated by $p.$ Therefore the image of the homomorphism $z\mapsto |z|$ is a discrete subgroup of $\mathbb {R} _{+},$ generated by $p.$ Since this group is non-trivial, it is generated by $Q=p^{m}$ for some $m\in \mathbb {N} .$ Choose $z_{1}\in I_{K},$ so that $|z_{1}|=Q,$ then $I_{K}$ is the direct product of $I_{K}^{1}$ and the subgroup generated by $z_{1}.$ This subgroup is discrete and isomorphic to $\mathbb {Z} .$

$\operatorname {char} (K)=0.$ For $\lambda \in \mathbb {R} _{+}$ define:

$z(\lambda )=(z_{v})_{v},\quad z_{v}={\begin{cases}1&v\notin P_{\infty }\\\lambda &v\in P_{\infty }\end{cases}}$

The map $\lambda \mapsto z(\lambda )$ is an isomorphism of $\mathbb {R} _{+}$ in a closed subgroup $M$ of $I_{K}$ and $I_{K}\cong M\times I_{K}^{1}.$ The isomorphism is given by multiplication:

${\begin{cases}\phi :M\times I_{K}^{1}\to I_{K},\\((\alpha _{v})_{v},(\beta _{v})_{v})\mapsto (\alpha _{v}\beta _{v})_{v}\end{cases}}$

Obviously, $\phi$ is a homomorphism. To show it is injective, let $(\alpha _{v}\beta _{v})_{v}=1.$ Since $\alpha _{v}=1$ for $v<\infty ,$ it stands that $\beta _{v}=1$ for $v<\infty .$ Moreover, it exists a $\lambda \in \mathbb {R} _{+},$ so that $\alpha _{v}=\lambda$ for $v|\infty .$ Therefore, $\beta _{v}=\lambda ^{-1}$ for $v|\infty .$ Moreover $\textstyle \prod _{v}|\beta _{v}|_{v}=1,$ implies $\lambda ^{n}=1,$ where $n$ is the number of infinite places of $K.$ As a consequence $\lambda =1$ and therefore $\phi$ is injective. To show surjectivity, let $\gamma =(\gamma _{v})_{v}\in I_{K}.$ We define $\lambda :=|\gamma |^{\frac {1}{n}}$ and furthermore, we define $\alpha _{v}=1$ for $v<\infty$ and $\alpha _{v}=\lambda$ for $v|\infty .$ Define $\textstyle \beta ={\frac {\gamma }{\alpha }}.$ It stands, that $\textstyle |\beta |={\frac {|\gamma |}{|\alpha |}}={\frac {\lambda ^{n}}{\lambda ^{n}}}=1.$ Therefore, $\phi$ is surjective.

The other equations follow similarly.

Characterisation of the idele group

Theorem. Let $K$ be a number field. There exists a finite set of places $S,$ such that:

$I_{K}=\left(I_{K,S}\times \prod _{v\notin S}O_{v}^{\times }\right)K^{\times }=\left(\prod _{v\in S}K_{v}^{\times }\times \prod _{v\notin S}O_{v}^{\times }\right)K^{\times }.$

Proof. The class number of a number field is finite so let ${\mathfrak {a}}_{1},\ldots ,{\mathfrak {a}}_{h}$ be the ideals, representing the classes in $\operatorname {Cl} _{K}.$ These ideals are generated by a finite number of prime ideals ${\mathfrak {p}}_{1},\ldots ,{\mathfrak {p}}_{n}.$ Let $S$ be a finite set of places containing $P_{\infty }$ and the finite places corresponding to ${\mathfrak {p}}_{1},\ldots ,{\mathfrak {p}}_{n}.$ Consider the isomorphism:

$I_{K}/\left(\prod _{v<\infty }O_{v}^{\times }\times \prod _{v|\infty }K_{v}^{\times }\right)\cong J_{K},$

induced by

$(\alpha _{v})_{v}\mapsto \prod _{v<\infty }{\mathfrak {p}}_{v}^{v(\alpha _{v})}.$

At infinite places the statement is obvious so we prove the statement for finite places. The inclusion ″$\supset$″ is obvious. Let $\alpha \in I_{K,{\text{fin}}}.$ The corresponding ideal $\textstyle (\alpha )=\prod _{v<\infty }{\mathfrak {p}}_{v}^{v(\alpha _{v})}$ belongs to a class ${\mathfrak {a}}_{i}K^{\times },$ meaning $(\alpha )={\mathfrak {a}}_{i}(a)$ for a principal ideal $(a).$ The idele $\alpha '=\alpha a^{-1}$ maps to the ideal ${\mathfrak {a}}_{i}$ under the map $I_{K,{\text{fin}}}\to J_{K}.$ That means $\textstyle {\mathfrak {a}}_{i}=\prod _{v<\infty }{\mathfrak {p}}_{v}^{v(\alpha '_{v})}.$ Since the prime ideals in ${\mathfrak {a}}_{i}$ are in $S,$ it follows $v(\alpha '_{v})=0$ for all $v\notin S,$ that means $\alpha '_{v}\in O_{v}^{\times }$ for all $v\notin S.$ It follows, that $\alpha '=\alpha a^{-1}\in I_{K,S},$ therefore $\alpha \in I_{K,S}K^{\times }.$

Applications

Finiteness of the class number of a number field

In the previous section we used the fact that the class number of a number field is finite. Here we would like to prove this statement:

Theorem (finiteness of the class number of a number field). Let $K$ be a number field. Then $|\operatorname {Cl} _{K}|<\infty .$

Proof. The map

${\begin{cases}I_{K}^{1}\to J_{K}\\\left((\alpha _{v})_{v<\infty },(\alpha _{v})_{v|\infty }\right)\mapsto \prod _{v<\infty }{\mathfrak {p}}_{v}^{v(\alpha _{v})}\end{cases}}$

is surjective and therefore $\operatorname {Cl} _{K}$ is the continuous image of the compact set $I_{K}^{1}/K^{\times }.$ Thus, $\operatorname {Cl} _{K}$ is compact. In addition it is discrete and so finite.

Remark. There is a similar result for the case of a global function field. In this case, the so-called divisor group is defined. It can be shown, that the quotient of the set of all divisors of degree $0$ by the set of the principal divisors is a finite group.

Group of units and Dirichlet's unit theorem

Let $P\supset P_{\infty }$ be a finite set of places. Define

${\begin{aligned}\Omega (P)&:=\prod _{v\in P}K_{v}^{\times }\times \prod _{v\notin P}O_{v}^{\times }=(\mathbb {A} _{K}(P))^{\times }\\E(P)&:=K^{\times }\cap \Omega (P)\end{aligned}}$

Then $E(P)$ is a subgroup of $K^{\times },$ containing all elements $\xi \in K^{\times }$ satisfying $v(\xi )=0$ for all $v\notin P.$ Since $K^{\times }$ is discrete in $I_{K},$ $E(P)$ is a discrete subgroup of $\Omega (P)$ and with the same argument, $E(P)$ is discrete in $\Omega _{1}(P):=\Omega (P)\cap I_{K}^{1}.$

An alternative definition is: $E(P)=K(P)^{\times },$ where $K(P)$ is a subring of $K$ defined by

$K(P):=K\cap \left(\prod _{v\in P}K_{v}\times \prod _{v\notin P}O_{v}\right).$

As a consequence, $K(P)$ contains all elements $\xi \in K,$ which fulfil $v(\xi )\geq 0$ for all $v\notin P.$

Lemma 1. Let $0<c\leq C<\infty .$ The following set is finite:

$\left\{\eta \in E(P):\left.{\begin{cases}|\eta _{v}|_{v}=1&\forall v\notin P\\c\leq |\eta _{v}|_{v}\leq C&\forall v\in P\end{cases}}\right\}\right\}.$

Proof. Define

$W:=\left\{(\alpha _{v})_{v}:\left.{\begin{cases}|\alpha _{v}|_{v}=1&\forall v\notin P\\c\leq |\alpha _{v}|_{v}\leq C&\forall v\in P\end{cases}}\right\}\right\}.$

$W$ is compact and the set described above is the intersection of $W$ with the discrete subgroup $K^{\times }$ in $I_{K}$ and therefore finite.

Lemma 2. Let $E$ be set of all $\xi \in K$ such that $|\xi |_{v}=1$ for all $v.$ Then $E=\mu (K),$ the group of all roots of unity of $K.$ In particular it is finite and cyclic.

Proof. All roots of unity of $K$ have absolute value $1$ so $\mu (K)\subset E.$ For converse note that Lemma 1 with $c=C=1$ and any $P$ implies $E$ is finite. Moreover $E\subset E(P)$ for each finite set of places $P\supset P_{\infty }.$ Finally Suppose there exists $\xi \in E,$ which is not a root of unity of $K.$ Then $\xi ^{n}\neq 1$ for all $n\in \mathbb {N}$ contradicting the finiteness of $E.$

Unit Theorem. $E(P)$ is the direct product of $E$ and a group isomorphic to $\mathbb {Z} ^{s},$ where $s=0,$ if $P=\emptyset$ and $s=|P|-1,$ if $P\neq \emptyset .$

Dirichlet's Unit Theorem. Let $K$ be a number field. Then $O^{\times }\cong \mu (K)\times \mathbb {Z} ^{r+s-1},$ where $\mu (K)$ is the finite cyclic group of all roots of unity of $K,r$ is the number of real embeddings of $K$ and $s$ is the number of conjugate pairs of complex embeddings of $K.$ It stands, that $[K:\mathbb {Q} ]=r+2s.$

Remark. The Unit Theorem is a generalisation of Dirichlet's Unit Theorem. To see this let $K$ be a number field. We already know that $E=\mu (K),$ set $P=P_{\infty }$ and note $|P_{\infty }|=r+s.$ Then we have:

${\begin{aligned}E\times \mathbb {Z} ^{r+s-1}=E(P_{\infty })&=K^{\times }\cap \left(\prod _{v|\infty }K_{v}^{\times }\times \prod _{v<\infty }O_{v}^{\times }\right)\\&\cong K^{\times }\cap \left(\prod _{v<\infty }O_{v}^{\times }\right)\\&\cong O^{\times }\end{aligned}}$

Approximation theorems

Weak Approximation Theorem. Let $|\cdot |_{1},\ldots ,|\cdot |_{N},$ be inequivalent valuations of $K.$ Let $K_{n}$ be the completion of $K$ with respect to $|\cdot |_{n}.$ Embed $K$ diagonally in $K_{1}\times \cdots \times K_{N}.$ Then $K$ is everywhere dense in $K_{1}\times \cdots \times K_{N}.$ In other words, for each $\varepsilon >0$ and for each $(\alpha _{1},\ldots ,\alpha _{N})\in K_{1}\times \cdots \times K_{N},$ there exists $\xi \in K,$ such that:

$\forall n\in \{1,\ldots ,N\}:\quad |\alpha _{n}-\xi |_{n}<\varepsilon .$

Strong Approximation Theorem. Let $v_{0}$ be a place of $K.$ Define

$V:={\prod _{v\neq v_{0}}}^{'}K_{v}.$

Then $K$ is dense in $V.$

Remark. The global field is discrete in its adele ring. The strong approximation theorem tells us that, if we omit one place (or more), the property of discreteness of $K$ is turned into a denseness of $K.$

Hasse principle

Hasse-Minkowski Theorem. A quadratic form on $K$ is zero, if and only if, the quadratic form is zero in each completion $K_{v}.$

Remark. This is the Hasse principle for quadratic forms. For polynomials of degree larger than 2 the Hasse principle isn't valid in general. The idea of the Hasse principle (also known as local–global principle) is to solve a given problem of a number field $K$ by doing so in its completions $K_{v}$ and then concluding on a solution in $K.$

Characters on the adele ring

Definition. Let $G$ be a locally compact abelian group. The character group of $G$ is the set of all characters of $G$ and is denoted by ${\widehat {G}}.$ Equivalently ${\widehat {G}}$ is the set of all continuous group homomorphisms from $G$ to $\mathbb {T} :=\{z\in \mathbb {C} :|z|=1\}.$ We equip ${\widehat {G}}$ with the topology of uniform convergence on compact subsets of $G.$ One can show that ${\widehat {G}}$ is also a locally compact abelian group.

Theorem. The adele ring is self-dual: $\mathbb {A} _{K}\cong {\widehat {\mathbb {A} _{K}}}.$

Proof. By reduction to local coordinates it is sufficient to show each $K_{v}$ is self-dual. This can done by using a fixed character of $K_{v}.$ We illustrate this idea by showing $\mathbb {R}$ is self-dual. Define:

${\begin{cases}e_{\infty }:\mathbb {R} \to \mathbb {T} \\e_{\infty }(t):=\exp(2\pi it)\end{cases}}$

Then the following map is an isomorphism which respects topologies:

${\begin{cases}\phi :\mathbb {R} \to {\widehat {\mathbb {R} }}\\s\mapsto {\begin{cases}\phi _{s}:\mathbb {R} \to \mathbb {T} \\\phi _{s}(t):=e_{\infty }(ts)\end{cases}}\end{cases}}$

Theorem (algebraic and continuous duals of the adele ring). Let $\chi$ be a non-trivial character of $\mathbb {A} _{K},$ which is trivial on $K.$ Let $E$ be a finite-dimensional vector-space over $K.$ Let $E^{\star }$ and $\mathbb {A} _{E}^{\star }$ be the algebraic duals of $E$ and $\mathbb {A} _{E}.$ Denote the topological dual of $\mathbb {A} _{E}$ by $\mathbb {A} _{E}'$ and use $\langle \cdot ,\cdot \rangle$ and $[{\cdot },{\cdot }]$ to indicate the natural bilinear pairings on $\mathbb {A} _{E}\times \mathbb {A} _{E}'$ and $\mathbb {A} _{E}\times \mathbb {A} _{E}^{\star }.$ Then the formula $\langle e,e'\rangle =\chi ([e,e^{\star }])$ for all $e\in \mathbb {A} _{E}$ determines an isomorphism $e^{\star }\mapsto e'$ of $\mathbb {A} _{E}^{\star }$ onto $\mathbb {A} _{E}',$ where $e'\in \mathbb {A} _{E}'$ and $e^{\star }\in \mathbb {A} _{E}^{\star }.$ Moreover, if $e^{\star }\in \mathbb {A} _{E}^{\star }$ fulfils $\chi ([e,e^{\star }])=1$ for all $e\in E,$ then $e^{\star }\in E^{\star }.$

Tate's thesis

With the help of the characters of $\mathbb {A} _{K},$ we can do Fourier analysis on the adele ring. John Tate in his thesis "Fourier analysis in number fields and Heckes Zeta functions" proved results about Dirichlet L-functions using Fourier analysis on the adele ring and the idele group. Therefore, the adele ring and the idele group have been applied to study the Riemann zeta function and more general zeta functions and the L-functions. We can define adelic forms of these functions and we can represent them as integrals over the adele ring or the idele group, with respect to corresponding Haar measures. We can show functional equations and meromorphic continuations of these functions. For example, for all $s\in \mathbb {C}$ with $\Re (s)>1,$

$\int _{\widehat {\mathbb {Z} }}|x|^{s}d^{\times }x=\zeta (s),$

where $d^{\times }x$ is the unique Haar measure on $I_{\mathbb {Q} ,{\text{fin}}}$ normalized such that ${\widehat {\mathbb {Z} }}^{\times }$ has volume one and is extended by zero to the finite adele ring. As a result the Riemann zeta function can be written as an integral over (a subset of) the adele ring.

Automorphic forms

The theory of automorphic forms is a generalization of Tate's thesis by replacing the idele group with analogous higher dimensional groups. To see this note:

${\begin{aligned}I_{\mathbb {Q} }&=\operatorname {GL} (1,\mathbb {A} _{\mathbb {Q} })\\I_{\mathbb {Q} }^{1}&=(\operatorname {GL} (1,\mathbb {A} _{\mathbb {Q} }))^{1}:=\{x\in \operatorname {GL} (1,\mathbb {A} _{\mathbb {Q} }):|x|=1\}\\\mathbb {Q} ^{\times }&=\operatorname {GL} (1,\mathbb {Q} )\end{aligned}}$

Based on these identification a natural generalization would be to replace the idele group and the 1-idele with:

${\begin{aligned}I_{\mathbb {Q} }&\leftrightsquigarrow \operatorname {GL} (2,\mathbb {A} _{\mathbb {Q} })\\I_{\mathbb {Q} }^{1}&\leftrightsquigarrow (\operatorname {GL} (2,\mathbb {A} _{\mathbb {Q} }))^{1}:=\{x\in \operatorname {GL} (2,\mathbb {A} _{\mathbb {Q} }):|\det(x)|=1\}\\\mathbb {Q} &\leftrightsquigarrow \operatorname {GL} (2,\mathbb {Q} )\end{aligned}}$

And finally

$\mathbb {Q} ^{\times }\backslash I_{\mathbb {Q} }^{1}\cong \mathbb {Q} ^{\times }\backslash I_{\mathbb {Q} }\leftrightsquigarrow (\operatorname {GL} (2,\mathbb {Q} )\backslash (\operatorname {GL} (2,\mathbb {A} _{\mathbb {Q} }))^{1}\cong (\operatorname {GL} (2,\mathbb {Q} )Z_{\mathbb {R} })\backslash \operatorname {GL} (2,\mathbb {A} _{\mathbb {Q} }),$

where $Z_{\mathbb {R} }$ is the centre of $\operatorname {GL} (2,\mathbb {R} ).$ Then we define an automorphic form as an element of $L^{2}((\operatorname {GL} (2,\mathbb {Q} )Z_{\mathbb {R} })\backslash \operatorname {GL} (2,\mathbb {A} _{\mathbb {Q} })).$ In other words an automorphic form is a functions on $\operatorname {GL} (2,\mathbb {A} _{\mathbb {Q} })$ satisfying certain algebraic and analytic conditions. For studying automorphic forms, it is important to know the representations of the group $\operatorname {GL} (2,\mathbb {A} _{\mathbb {Q} }).$ It is also possible to study automorphic L-functions, which can be described as integrals over $\operatorname {GL} (2,\mathbb {A} _{\mathbb {Q} }).$

We could generalize even further by replacing $\mathbb {Q}$ with a number field and $\operatorname {GL} (2)$ with an arbitrary reductive algebraic group.

Further applications

A generalisation of Artin reciprocity law leads to the connection of representations of $\operatorname {GL} (2,\mathbb {A} _{K})$ and of Galois representations of $K$ (Langlands program).

The idele class group is a key object of class field theory, which describes abelian extensions of the field. The product of the local reciprocity maps in local class field theory gives a homomorphism of the idele group to the Galois group of the maximal abelian extension of the global field. The Artin reciprocity law, which is a high level generalisation of the Gauss quadratic reciprocity law, states that the product vanishes on the multiplicative group of the number field. Thus, we obtain the global reciprocity map of the idele class group to the abelian part of the absolute Galois group of the field.

The self-duality of the adele ring of the function field of a curve over a finite field easily implies the Riemann–Roch theorem and the duality theory for the curve.

Sources

• Cassels, John; Fröhlich, Albrecht (1967). Algebraic number theory: proceedings of an instructional conference, organized by the London Mathematical Society, (a NATO Advanced Study Institute). Vol. XVIII. London: Academic Press. ISBN 978-0-12-163251-9. {{cite book}}: Invalid |ref=harv (help) 366 pages.
• Neukirch, Jürgen (2007). Algebraische Zahlentheorie, unveränd. nachdruck der 1. aufl. edn (in German). Vol. XIII. Berlin: Springer. ISBN 9783540375470. {{cite book}}: Invalid |ref=harv (help) 595 pages.
• Weil, André (1967). Basic number theory. Vol. XVIII. Berlin; Heidelberg; New York: Springer. ISBN 978-3-662-00048-9. {{cite book}}: Invalid |ref=harv (help) 294 pages.
• Deitmar, Anton (2010). Automorphe Formen (in German). Vol. VIII. Berlin; Heidelberg (u.a.): Springer. ISBN 978-3-642-12389-4. {{cite book}}: Invalid |ref=harv (help) 250 pages.
• Lang, Serge (1994). Algebraic number theory, Graduate Texts in Mathematics 110 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-94225-4. {{cite book}}: Invalid |ref=harv (help)