Idele group

In number theory, the idele group is a way of packaging the multiplicative arithmetic of a global field at all of its completions at once, so that it contains the information of unique factorization as well as the data relating to units. Formally, the idele group of a global field K is the restricted direct product \mathbb A_K^\times = \prod_v' K_v^\times of the multiplicative groups of the completions of K, taken with respect to the unit groups \mathcal O_v^\times at the non-archimedean places. Equivalently, it is the group of invertible elements of the adele ring \mathbb A_K, equipped with a topology finer than the subspace topology inherited from \mathbb A_K.

The quotient C_K=\mathbb A_K^\times/K^\times is the idele class group. Ideles and idele class groups are used in class field theory. They were exploited by John Tate in his thesis to formulate global zeta and L-functions and Hecke characters.

Definition

Let K be a global field, and let v run over the places of K. For each place v, let K_v denote the completion of K at v. If v is non-archimedean, let \mathcal O_v be the corresponding valuation ring and let \mathcal O_v^\times be its group of units.

The idele group of K, usually denoted \mathbb A_K^\times or I_K, is the restricted product

\mathbb A_K^\times = \prod_v' K_v^\times

of the groups K_v^\times, taken with respect to the subgroups \mathcal O_v^\times at the non-archimedean places. Thus an idele is a family

x=(x_v)_v,\qquad x_v\in K_v^\times,

such that

x_v\in \mathcal O_v^\times

for all but finitely many non-archimedean places v. Multiplication is defined componentwise.^[1][2]

Equivalently, the idele group is the group of invertible elements of the adele ring \mathbb A_K. However, its topology is not the subspace topology inherited from \mathbb A_K; it is the restricted product topology, or equivalently the topology induced by the embedding

\mathbb A_K^\times \longrightarrow \mathbb A_K \times \mathbb A_K,\qquad x\mapsto (x,x^{-1}).

The multiplicative group K^\times embeds diagonally in \mathbb A_K^\times. The quotient

C_K=\mathbb A_K^\times/K^\times

is called the idele class group of K.

Motivation

The idele group may be viewed as a topological refinement of the group of fractional ideals of a number field. If K is a number field with ring of integers \mathcal O_K, every nonzero fractional ideal has a unique factorization

\mathfrak a=\prod_{\mathfrak p}\mathfrak p^{n_{\mathfrak p}},

where \mathfrak p runs over the nonzero prime ideals of \mathcal O_K and all but finitely many integers n_{\mathfrak p} are zero. Thus the group of fractional ideals records, for each finite place of K, an integral valuation.

An idele records similar local valuation data, but with additional local information. For an idele x=(x_v)_v, the component x_{\mathfrak p}\in K_{\mathfrak p}^{\times} at a finite place determines an integer v_{\mathfrak p}(x_{\mathfrak p}). Since x_{\mathfrak p} is a unit for all but finitely many \mathfrak p, these integers define a fractional ideal

(x)_{\mathrm{fin}}=\prod_{\mathfrak p}\mathfrak p^{v_{\mathfrak p}(x_{\mathfrak p})}.

This gives a surjective homomorphism from the idele group to the group of fractional ideals. The diagonal embedding K^\times\hookrightarrow \mathbb A_K^\times sends an element a\in K^\times to the principal idele whose associated fractional ideal is the principal ideal (a). Consequently, passing to quotients gives a natural surjection

\mathbb A_K^\times/K^\times \longrightarrow \operatorname{Cl}(K),

from the idele class group to the ordinary ideal class group.^[3]

Thus the idele class group enlarges the ideal class group. It extends the finite-prime data measured by fractional ideals with the unit groups at finite places and the multiplicative groups at the archimedean places. This additional topological information is important in class field theory and in the theory of Hecke characters, where characters of the idele class group replace characters defined only on ideal class groups or ray class groups.

Topology and Haar measure

Although the idele group \mathbb A_K^\times is the group of invertible elements of the adele ring \mathbb A_K, it is not usually equipped with the subspace topology inherited from \mathbb A_K. With the subspace topology, inversion need not be continuous. Instead, \mathbb A_K^\times is given the restricted product topology

\mathbb A_K^\times=\prod_v' K_v^\times,

where the restricted product is taken with respect to the compact open subgroups \mathcal O_v^\times at the non-archimedean places. A basis of open neighbourhoods of the identity is given by products

\prod_v U_v,

where U_v is an open neighbourhood of 1 in K_v^\times and U_v=\mathcal O_v^\times for all but finitely many non-archimedean places v. Equivalently, this is the topology induced by the embedding

\mathbb A_K^\times \longrightarrow \mathbb A_K\times \mathbb A_K,\qquad x\mapsto (x,x^{-1}).

With this topology, \mathbb A_K^\times is a locally compact topological group.^[2][1]

Since the idele group is locally compact, it has a Haar measure, usually denoted d^\times x. This measure is obtained as a product of local multiplicative Haar measures on the groups K_v^\times. At a non-archimedean place v, the local measure is commonly normalized so that

\operatorname{vol}(\mathcal O_v^\times)=1.

At the real place, a standard multiplicative Haar measure on \mathbb R^\times is

d^\times x=\frac{dx}{|x|},

up to multiplication by a positive constant; analogous normalizations are used at complex places. These local choices combine to give a multiplicative Haar measure on \mathbb A_K^\times. Such measures are used in harmonic analysis on the ideles, especially in Tate's thesis and in the analytic theory of Hecke L-functions.^[4][5]

Norm map and norm-one ideles

The idele group carries a homomorphism, usually called the idele norm or module into the positive reals. Choose the standard normalized absolute value |\cdot|_v on each completion K_v: for a non-archimedean place v, it is normalized so that

|\varpi_v|_v=q_v^{-1},

where \varpi_v is a uniformizer and q_v is the size of the residue field. At the archimedean places one uses the usual normalized absolute values, with the complex absolute value taken squared. For an idele x=(x_v)_v, define

|x|_{\mathbb A}=\prod_v |x_v|_v.

This product is finite, since x_v\in\mathcal O_v^\times for all but finitely many non-archimedean places, and hence |x_v|_v=1 for all but finitely many v. Thus

|\cdot|_{\mathbb A}:\mathbb A_K^\times\to \mathbb R_{>0}

is a continuous group homomorphism.^[1][2]

The norm-one ideles are the elements in the kernel of this homomorphism:

\mathbb A_K^1=\{x\in\mathbb A_K^\times: |x|_{\mathbb A}=1\}.

By the product formula for global fields, every element of K^\times, embedded diagonally in \mathbb A_K^\times, has idele norm one. Hence

K^\times\subset \mathbb A_K^1.

The quotient

C_K^1=\mathbb A_K^1/K^\times

is called the group of norm-one idele classes. It is a compact group.^[2][6]

The idele norm descends to a homomorphism on the idele class group,

|\cdot|_{\mathbb A}:C_K=\mathbb A_K^\times/K^\times\to \mathbb R_{>0},

whose kernel is C_K^1. For number fields this gives an exact sequence

1\longrightarrow C_K^1\longrightarrow C_K
 \xrightarrow{|\cdot|_{\mathbb A}} \mathbb R_{>0}\longrightarrow 1.

Thus the idele class group is not compact in the number field case, but its norm-one subgroup modulo K^\times is compact. This compactness is one of the idelic forms of the finiteness of the ideal class group together with the structure theorem for units.^[1][2]

For number fields, the idele norm is surjective onto \mathbb R_{>0}, and the above exact sequence splits after choosing a positive archimedean component. Thus C_K is, non-canonically or after such a choice, a product of the compact group C_K^1 with \mathbb R_{>0}. For global function fields, the image of the idele norm is instead a discrete subgroup of \mathbb R_{>0}, so the corresponding quotient is discrete and isomorphic to an infinite cyclic group.^[2][1]

Norms for field extensions

Let L/K be a finite extension of global fields. For each place v of K and each place w of L lying above v, there is a local norm map

N_{L_w/K_v}:L_w^\times\longrightarrow K_v^\times .

These local norm maps combine to give a continuous homomorphism on idele groups

N_{L/K}:\mathbb A_L^\times\longrightarrow \mathbb A_K^\times .

If y=(y_w)_w\in \mathbb A_L^\times, then the v-component of N_{L/K}(y) is

\left(N_{L/K}(y)\right)_v = \prod_{w\mid v} N_{L_w/K_v}(y_w).

This product is finite for each fixed v. Moreover, for all but finitely many non-archimedean places w, the component y_w lies in \mathcal O_w^\times, and its local norm lies in \mathcal O_v^\times. Hence N_{L/K}(y) is again an idele of K. The continuity follows from the continuity of the local norm maps and from the restricted product topology.^[1][2]

The norm map is compatible with principal ideles. If a\in L^\times is embedded diagonally in \mathbb A_L^\times, then

N_{L/K}(a)

is the principal idele of K associated with the field norm N_{L/K}(a)\in K^\times. Consequently, the idele norm descends to a continuous homomorphism on idele class groups,

N_{L/K}:C_L\longrightarrow C_K,

where C_L=\mathbb A_L^\times/L^\times and C_K=\mathbb A_K^\times/K^\times.

The embedding of K into L also gives a natural homomorphism

\mathbb A_K^\times\longrightarrow \mathbb A_L^\times .

Explicitly, an idele x=(x_v)_v of K is sent to the idele whose component at w\mid v is the image of x_v in L_w^\times. Under this embedding,

N_{L/K}(x)=x^{[L:K]},

where the power is taken componentwise. This follows from the identity

\prod_{w\mid v} N_{L_w/K_v}(x_v) = x_v^{\sum_{w\mid v}[L_w:K_v]} = x_v^{[L:K]}.

The field-extension norm should be distinguished from the idele norm or module |x|_{\mathbb A}. They are nevertheless compatible: with the standard normalized absolute values,

|N_{L/K}(y)|_{\mathbb A_K} = |y|_{\mathbb A_L}.

In particular, N_{L/K} maps the norm-one idele group \mathbb A_L^1 into \mathbb A_K^1 and induces a homomorphism

C_L^1\longrightarrow C_K^1.

In global class field theory, the image N_{L/K}(C_L) is called the norm subgroup of C_K. For a finite abelian extension L/K, the global Artin reciprocity map identifies the quotient

C_K/N_{L/K}(C_L)

with the Galois group \operatorname{Gal}(L/K), up to the usual convention concerning arithmetic or geometric Frobenius.^[3][6][7]

Example: the rational numbers

For K=\mathbb Q, the finite adele ring is

\mathbb A_{\mathbb Q,\mathrm{fin}} = \prod_p' \mathbb Q_p,

and the finite integral adeles are

\widehat{\mathbb Z}=\prod_p \mathbb Z_p.

The finite ideles are

\mathbb A_{\mathbb Q,\mathrm{fin}}^\times = \prod_p' \mathbb Q_p^\times,

where the restricted product is taken with respect to \mathbb Z_p^\times. The idele group of \mathbb Q is

\mathbb A_{\mathbb Q}^\times = \mathbb A_{\mathbb Q,\mathrm{fin}}^\times \times \mathbb R^\times .

Every idele class has a representative of the form

(u,t)\in \widehat{\mathbb Z}^{\times}\times \mathbb R_{>0}.

Indeed, multiplying by a rational number changes the finite valuations and can be used to make all finite components p-adic units; the remaining positive real factor records the idele norm. Thus

\mathbb A_{\mathbb Q}^\times/\mathbb Q^\times \cong \widehat{\mathbb Z}^{\times}\times \mathbb R_{>0}.

Similarly, the norm-one idele classes are

\mathbb A_{\mathbb Q}^{1}/\mathbb Q^\times \cong \widehat{\mathbb Z}^{\times}.

This reflects the fact that \mathbb Q has trivial ideal class group: the remaining finite part of the idele class group comes from the local unit groups \mathbb Z_p^\times.

Class field theory

The idele class group yields a formulation of class field theory. Global class field theory describes the abelian extensions of a global field K in terms of topological quotients of C_K=\mathbb A_K^\times/K^\times.

The main result is the global Artin reciprocity law. In one formulation, for every finite abelian extension L/K there is a canonical reciprocity homomorphism

\theta_{L/K}: C_K \longrightarrow \operatorname{Gal}(L/K),

whose kernel is the norm subgroup N_{L/K}(C_L)\subset C_K.

The reciprocity homomorphism induces an isomorphism C_K/N_{L/K}(C_L)\cong \operatorname{Gal}(L/K), up to a conventional choice of arithmetic or geometric Frobenius automorphism.^[3][6][7]

Thus, finite abelian extensions of K correspond to open subgroups of finite index in the idele class group. Under this correspondence, an extension L/K is associated with the subgroup N_{L/K}(C_L). This replaces the older formulation of class field theory in terms of ideal class groups, ray class groups, and congruence conditions by a topological statement about quotients of C_K.^[3][7]

The idelic formulation also incorporates the local reciprocity maps of local class field theory: for each place v of K, local class field theory relates K_v^\times to the abelianized Galois group of K_v. The global reciprocity map is compatible with these local maps through the embedding of each local multiplicative group into the idele group. At an unramified finite place, a uniformizer maps to a Frobenius element, with the precise inverse depending on the convention used for the Artin map.^[3][7]

Classical ideal-theoretic class field theory is then a special case. Quotients of the idele class group by certain subgroups recover ray class groups, and the corresponding abelian extensions are the ray class fields. In particular, the Hilbert class field is obtained from the quotient associated with the ordinary ideal class group, packaging the relation between ideles, fractional ideals, and ideal classes.^[3][6]

For the maximal abelian extension K^{\mathrm{ab}}, the finite-level reciprocity maps are compatible as L varies over finite abelian extensions of K. They combine into a global reciprocity map from the idele class group to \operatorname{Gal}(K^{\mathrm{ab}}/K). Thus the abelianized absolute Galois group of K is described by the system of finite quotients of the idele class group.^[3][7]

Hecke characters and L-functions

A Hecke character of a global field K can be described as a continuous homomorphism

\chi:\mathbb A_K^\times/K^\times \to \mathbb C^\times,

or equivalently as a continuous character of the idele group \mathbb A_K^\times that is trivial on the diagonally embedded subgroup K^\times. Such characters are the automorphic characters of \operatorname{GL}_1(\mathbb A_K).

Writing an idele as x=(x_v)_v, a Hecke character decomposes into local characters

\chi_v:K_v^\times\to \mathbb C^\times,

with

\chi(x)=\prod_v \chi_v(x_v).

For all but finitely many non-archimedean places v, the local character \chi_v is unramified, meaning that it is trivial on \mathcal O_v^\times. At such a place its value is determined by \chi_v(\varpi_v), where \varpi_v is a uniformizer of K_v.

Associated to a Hecke character is a global L-function, defined for suitable s by an Euler product

L(s,\chi)=\prod_v L_v(s,\chi_v).

At an unramified non-archimedean place v, the local factor has the form

L_v(s,\chi_v)=\left(1-\chi_v(\varpi_v)q_v^{-s}\right)^{-1},

where q_v is the size of the residue field. The remaining finitely many finite places give ramified local factors, and the archimedean places contribute gamma factors. These local factors combine to form the completed Hecke L-function.^[4][8]

Classical Dirichlet characters and ideal class characters occur as special cases. For example, over \mathbb Q, Dirichlet characters can be interpreted as finite-order Hecke characters with prescribed finite conductors. More generally, ray class characters of a number field can be realized as finite-order characters of quotients of the idele class group.

Hecke L-functions are among the basic examples of automorphic L-functions. In Tate's thesis, the analytic continuation and functional equation of these L-functions are obtained by harmonic analysis on the adele ring and the idele group. This approach recovers the analytic theory of Dirichlet L-functions and Hecke's original L-series, while also explaining their local-global factorization in terms of the product structure of the ideles.^[4][9]

Relation with the ideal class group

For a number field K, the idele group refines the ordinary ideal-theoretic arithmetic of K. Let \mathcal O_K be the ring of integers of K, let J_K be the group of nonzero fractional ideals of K, and let

\widehat{\mathcal O}_K=\prod_{\mathfrak p}\mathcal O_{\mathfrak p}

be the profinite completion of \mathcal O_K, where \mathfrak p runs over the nonzero prime ideals of \mathcal O_K. Its group of units is

\widehat{\mathcal O}_K^\times=\prod_{\mathfrak p}\mathcal O_{\mathfrak p}^{\times}.

Let I_{K,\mathrm{fin}} denote the finite idele group,

I_{K,\mathrm{fin}}=\prod_{\mathfrak p}' K_{\mathfrak p}^{\times}.

There is a natural surjective homomorphism

I_{K,\mathrm{fin}}\longrightarrow J_K

defined by

x=(x_{\mathfrak p})_{\mathfrak p} \longmapsto \prod_{\mathfrak p}\mathfrak p^{v_{\mathfrak p}(x_{\mathfrak p})},

where v_{\mathfrak p} is the normalized additive valuation at \mathfrak p. The product is finite because x_{\mathfrak p}\in \mathcal O_{\mathfrak p}^{\times} for all but finitely many \mathfrak p. The kernel of this homomorphism is exactly \widehat{\mathcal O}_K^\times. Hence

I_{K,\mathrm{fin}}/\widehat{\mathcal O}_K^\times\cong J_K.

This identifies the group of fractional ideals with the quotient of the finite idele group obtained by forgetting the local unit components.^[1][6]

The diagonal embedding K^\times\hookrightarrow I_{K,\mathrm{fin}} is compatible with principal ideals. If a\in K^\times, then the finite idele whose components are all equal to a maps to the principal fractional ideal (a). Therefore the preceding homomorphism descends to a quotient map from finite idele classes to ideal classes. In particular,

\operatorname{Cl}(K) \cong I_{K,\mathrm{fin}}/K^\times\widehat{\mathcal O}_K^\times.

Equivalently, using the full idele group,

\operatorname{Cl}(K) \cong \mathbb A_K^\times \big/ K^\times \left( \widehat{\mathcal O}_K^\times \times \prod_{v\mid\infty}K_v^\times \right).

Thus the ordinary ideal class group is obtained from the idele class group by quotienting out the finite local unit groups and the archimedean multiplicative factors.

The same construction gives a useful way to view why ideles contain more information than ideals. Passing from an idele x=(x_v)_v to the associated fractional ideal records only the valuations v_{\mathfrak p}(x_{\mathfrak p}) at the finite places. It discards the unit components in \mathcal O_{\mathfrak p}^{\times} and also discards the archimedean components. These extra local and topological data are precisely what make the idele class group suitable for class field theory and for the theory of Hecke characters.

A proof sketch is as follows. For each finite prime \mathfrak p, choose a uniformizer \varpi_{\mathfrak p} of K_{\mathfrak p}. Every element of K_{\mathfrak p}^{\times} can be written as \varpi_{\mathfrak p}^n u, with n\in\mathbb Z and u\in\mathcal O_{\mathfrak p}^{\times}. Hence the valuation map records exactly the exponent of \mathfrak p. Since an idele is a unit at almost all finite places, only finitely many exponents are nonzero, so the formula above defines a fractional ideal. The kernel consists exactly of those finite ideles with all valuations zero, namely \widehat{\mathcal O}_K^\times. Surjectivity follows because any fractional ideal \prod_{\mathfrak p}\mathfrak p^{n_{\mathfrak p}} is represented by the finite idele whose \mathfrak p-component is \varpi_{\mathfrak p}^{n_{\mathfrak p}} for the finitely many primes appearing in the product and is 1 elsewhere. Finally, quotienting by the diagonal image of K^\times identifies principal fractional ideals with principal ideles, giving the ideal class group.

Further structure and proof sketches

The following standard structural facts give equivalent descriptions of the idele topology, related subgroups, and some compactness and decomposition results used in the arithmetic theory of ideles.

Topology induced from the adele ring

The topology on \mathbb A_K^\times can be described by a general construction for unit groups of topological rings. 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 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. It is necessary to show that 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 same condition applied to U^{-1}. The idele group is equipped with this topology.

The subset topology inherited from \mathbb A_K is not a suitable candidate in general, since the group of units of a topological ring equipped with the subset topology may not be a topological group. For example, the inverse map in \mathbb A_{\mathbb Q} is not continuous. The sequence

\begin{align} x_1&=(2,1,\ldots)\\ x_2&=(1,3,1,\ldots)\\ x_3&=(1,1,5,1,\ldots)\\ &\vdots \end{align}

converges to 1\in \mathbb A_{\mathbb Q}. To see this, let U be a neighbourhood of 0; without loss of generality it can be assumed that

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, it follows that x_n-1\in U for n large enough. However, the inverses of this sequence do not converge to 1 in \mathbb A_{\mathbb Q}.

Subgroups attached to sets of places

For S a subset of places of K, set

I_{K,S}:=\mathbb A_{K,S}^{\times}, \qquad I_K^S:=(\mathbb A_K^S)^{\times}.

The following identities of topological groups hold:

\begin{align} 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{align}

Here the restricted product has the restricted product topology, generated by restricted open rectangles of the form

\prod_{v\in E}U_v\times \prod_{v\notin E}\mathcal 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.

It suffices to prove the identity for I_K; the other two follow similarly. First show the two sets are equal:

\begin{align} 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_vy_v=1\quad \forall v\}\\ &=\{x=(x_v)_v:x_v\in K_v^\times\ \forall v \text{ and }x_v\in\mathcal O_v^\times\text{ for almost all }v\}\\ &={\prod_v}'K_v^\times . \end{align}

In going from the second line to the third, x as well as x^{-1}=y have to be in \mathbb A_K, meaning x_v\in\mathcal O_v for almost all v and x_v^{-1}\in\mathcal O_v for almost all v. Therefore x_v\in\mathcal O_v^\times for almost all v.

Now the topology on the left-hand side equals the topology on the right-hand side. Every open restricted rectangle is open in the topology of the idele group. Conversely, for a given U\subset I_K open in the topology of the idele group, meaning that U\times U^{-1}\subset \mathbb A_K\times\mathbb A_K is open, for each u\in U there exists an open restricted rectangle contained in U and containing u. Therefore U is the union of all these restricted open rectangles and is open in the restricted product topology.

For each set of places S, I_{K,S} is a locally compact topological group. The local compactness follows from the description of I_{K,S} as a restricted product, and the topological group property follows from the preceding discussion on the group of units of a topological ring.

A neighbourhood system of 1\in I_K is given by all sets of the form

\prod_v U_v,

where U_v is a neighbourhood of 1\in K_v^\times and U_v=\mathcal O_v^\times for almost all v.

Finite extensions

Let L/K be a finite extension. Then

I_L={\prod_w}' L_w^\times,

where the restricted product is with respect to the unit groups \mathcal O_w^\times.

There is a canonical embedding of I_K in I_L. 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\mid 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 its components satisfy the following properties: a_w\in K_v^\times for w\mid v, and a_w=a_{w'} for w\mid v and w'\mid v over the same place v of K.

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}

Principal ideles and discreteness

There is a natural embedding of K^\times into I_K given by the diagonal map

a\mapsto (a,a,a,\ldots).

Since K^\times is a subset of K_v^\times for all v, the embedding is well-defined and injective. In analogy to the ideal class group, the elements of K^\times in I_K are called principal ideles.

The subgroup K^\times is closed and discrete in I_K. Therefore

C_K=I_K/K^\times

is a locally compact topological group and a Hausdorff space.

More generally, in the adelic algebra setting described below, A^\times is a discrete subgroup of \mathbb A_A^\times.

Product formula and compactness of norm-one idele classes

For \alpha=(\alpha_v)_v\in I_K, define

|\alpha|:=\prod_v|\alpha_v|_v.

Since \alpha is an idele, this product is finite and therefore well-defined. The set of norm-one ideles is

I_K^1:=\{x\in I_K:|x|=1\}=\ker(|\cdot|).

The subgroup I_K^1 is a closed subgroup of I_K. The \mathbb A_K-topology on I_K^1 equals the subspace topology of I_K on I_K^1.^[10]

The product formula states that

|k|=1

for all k\in K^\times.

Proof.

For number fields, the case of global function fields being similar, let K be a number field and a\in K^\times. It has to be shown that

\prod_v |a|_v=1.

For a finite place v for which the corresponding prime ideal \mathfrak p_v does not divide (a), v(a)=0 and therefore |a|_v=1. This is valid for almost all \mathfrak p_v. There is

\begin{align} \prod_v |a|_v &=\prod_{p\leq \infty}\prod_{v\mid p}|a|_v\\ &=\prod_{p\leq \infty}\prod_{v\mid p}|N_{K_v/\mathbb Q_p}(a)|_p\\ &=\prod_{p\leq \infty}|N_{K/\mathbb Q}(a)|_p . \end{align}

In going from the first line to the second, the identity

|a|_w=|N_{L_w/K_v}(a)|_v

is used, where v is a place of K and w is a place of L lying above v. Going from the second line to the third uses the compatibility of local and global norms. The norm is in \mathbb Q, so it remains to prove the product formula over \mathbb Q. Write

a=\pm \prod_{p<\infty}p^{v_p},

where v_p\in\mathbb Z is 0 for almost all p. Then

\begin{align} |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{align}

The following approximation lemma is used in the proof of compactness.

Lemma. There exists a constant C, depending only on K, such that for every \alpha=(\alpha_v)_v\in \mathbb A_K satisfying

\prod_v|\alpha_v|_v>C,

there exists \beta\in K^\times such that

|\beta|_v\leq |\alpha_v|_v

for all v.^[11]

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 that \delta_v=1 for almost all v. Then there exists \beta\in K^\times such that

|\beta|_v\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 \mathcal O_v. Define the adele \alpha=(\alpha_v)_v by \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

\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 such 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 be the constant of the lemma and suppose \alpha\in\mathbb A_K satisfies

\prod_v|\alpha_v|_v>C.

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. It can be claimed that 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. This proves the surjectivity of the natural projection. Since it is also continuous, compactness follows.^[2][12]

The rational numbers

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

\left(\prod_{p<\infty}|a_p|_p\right)\cdot 1=1.

Obviously \phi is a continuous group homomorphism. 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 it can be seen that q=1, which proves 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 there is

((\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,

it follows that \phi is surjective.

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

\begin{align} I_{\mathbb Q}&\cong I_{\mathbb Q}^1\times(0,\infty),\\ I_{\mathbb Q}^1&\cong I_{\mathbb Q,\mathrm{fin}}\times\{\pm1\}. \end{align}

The isomorphisms are given by

\begin{cases} \psi:I_{\mathbb Q}\to I_{\mathbb Q}^1\times(0,\infty),\\ a=(a_{\mathrm{fin}},a_\infty)\mapsto \left(a_{\mathrm{fin}},\frac{a_\infty}{|a|},|a|\right), \end{cases}

and

\begin{cases} \widetilde\psi:I_{\mathbb Q,\mathrm{fin}}\times\{\pm1\}\to I_{\mathbb Q}^1,\\ (a_{\mathrm{fin}},\varepsilon)\mapsto \left(a_{\mathrm{fin}},\frac{\varepsilon}{|a_{\mathrm{fin}}|}\right). \end{cases}

Decomposition of the idele group and idele class group

The idele norm gives the following decompositions:

\begin{align} 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_{>0}, &\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_{>0}, &\operatorname{char}(K)=0. \end{cases} \end{align}

Proof.

First suppose \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_{>0} generated by p. Therefore, for each z\in I_K, |z| is in the subgroup of \mathbb R_{>0} generated by p. Thus the image of the homomorphism z\mapsto |z| is a discrete subgroup of \mathbb R_{>0}. Since this group is nontrivial, it is generated by Q=p^m for some m\in\mathbb N. Choose z_1\in I_K such 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.

Now suppose \operatorname{char}(K)=0. For \lambda\in\mathbb R_{>0}, define

z(\lambda)=(z_v)_v, \qquad z_v= \begin{cases} 1,&v\nmid\infty,\\ \lambda,&v\mid\infty. \end{cases}

The map \lambda\mapsto z(\lambda) is an isomorphism of \mathbb R_{>0} onto 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\nmid\infty, it follows that \beta_v=1 for v\nmid\infty. Moreover, there exists a \lambda\in\mathbb R_{>0} such that \alpha_v=\lambda for v\mid\infty. Therefore \beta_v=\lambda^{-1} for v\mid\infty. Since

\prod_v|\beta_v|_v=1,

it follows that \lambda^n=1, where n is the number of archimedean places of K. Consequently \lambda=1, and therefore \phi is injective.

To show surjectivity, let \gamma=(\gamma_v)_v\in I_K. Define \lambda:=|\gamma|^{1/n}, and define \alpha_v=1 for v\nmid\infty and \alpha_v=\lambda for v\mid\infty. Let

\beta=\frac{\gamma}{\alpha}.

Then

|\beta|=\frac{|\gamma|}{|\alpha|}=\frac{\lambda^n}{\lambda^n}=1.

Therefore \phi is surjective. The statements for C_K follow similarly.

Characterisation by a finite set of places

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}\mathcal O_v^\times \right)K^\times = \left( \prod_{v\in S}K_v^\times \times \prod_{v\notin S}\mathcal 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 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 the archimedean places and the finite places corresponding to \mathfrak p_1,\ldots,\mathfrak p_n. Consider the isomorphism

I_K/ \left( \prod_{v<\infty}\mathcal O_v^\times \times \prod_{v\mid\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 immediate, so it remains to prove the statement for finite places. The inclusion \supset is obvious. Let \alpha\in I_{K,\mathrm{fin}}. The corresponding ideal

(\alpha)=\prod_{v<\infty}\mathfrak p_v^{v(\alpha_v)}

belongs to a class \mathfrak a_iK^\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,\mathrm{fin}}\to J_K. That means

\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 that v(\alpha'_v)=0 for all v\notin S. Thus \alpha'_v\in \mathcal O_v^\times for all v\notin S. It follows that \alpha'=\alpha a^{-1}\in I_{K,S}, and therefore \alpha\in I_{K,S}K^\times.

Ideles of finite-dimensional algebras

The construction also extends to finite-dimensional algebras over K. Let A be a finite-dimensional algebra over K. Since \mathbb A_A^\times is not a topological group with the subspace topology in general, equip \mathbb A_A^\times with the topology similar to I_K above and call \mathbb A_A^\times the idele group of A. The elements of the idele group are called ideles of A.^[2]

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 \mathcal O_v-module generated by \alpha in A_v. There exists a finite set of places P_0 containing the archimedean places such that for all v\notin P_0, \alpha_v is a compact subring of A_v. 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 onto its image in A_v\times A_v. For each v\notin P_0, the group \alpha_v^\times is an open and compact subgroup of A_v^\times.

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, and \mathbb A_A^\times is the union of all \mathbb A_A(P,\alpha)^\times. In the special case 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}\mathcal 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.

References

1. ^ Neukirch 1999, pp. Ch. VI, §1.
2. ^ Weil 1995, Ch. IV.
3. ^ Neukirch 1999, Ch. VI.
4. ^ Tate 1967.
5. ^ Ramakrishnan & Valenza 1999.
6. ^ Cassels & Fröhlich 1967.
7. ^ Weil 1995, Ch. VII.
8. ^ Weil 1995.
9. ^ Bump 1997.
10. ^ Cassels & Fröhlich 1967, p. 69.
11. ^ Cassels & Fröhlich 1967, p. 66.
12. ^ Cassels & Fröhlich 1967, p. 70.

• Neukirch, Jürgen (1999), Algebraic Number Theory, Vol. 322, Grundlehren der mathematischen Wissenschaften, Translated by Schappacher, Norbert, Springer, ISBN 978-3-540-65399-8.
• Weil, André (1995), Basic Number Theory, Classics in Mathematics, Springer, ISBN 978-3-540-58655-5.
• Cassels, J. W. S. & Fröhlich, Albrecht (eds.) (1967), "Algebraic Number Theory", London: Academic Press.
• Tate, John (1967), "Fourier analysis in number fields, and Hecke's zeta-functions", "Algebraic Number Theory", Cassels, J. W. S. & Fröhlich, Albrecht (eds.), London: Academic Press, pp. 305–347.
• Ramakrishnan, Dinakar & Valenza, Robert J. (1999), Fourier Analysis on Number Fields, Vol. 186, Graduate Texts in Mathematics, Springer, ISBN 978-0-387-98436-0.
• Bump, Daniel (1997), Automorphic Forms and Representations, Vol. 55, Cambridge Studies in Advanced Mathematics, Cambridge University Press, ISBN 978-0-521-65818-8.