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In mathematics, the adele ring[1] (also adelic ring or ring of adeles) 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,[2] 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.

Notation: Throughout this article, is a global field. That is, is an algebraic number field or a global function field. In the first case, is a finite field extension; in the second case is a finite field extension. We denote by a place (a representative of an equivalence class of valuations) of . The trivial valuation and the corresponding trivial value aren't allowed in the whole article. A finite/non-Archimedean valuation is written as or and an infinite/Archimedean valuation as We write for the finite set of all infinite places of and for a finite subset of all places of which contains In addition, we write for the completion of with respect to the valuation If the valuation is discrete, then we write for the valuation ring of We write for the maximal ideal of If this is a principal ideal, then we write for a uniformizing element. By fixing a suitable constant there is a one-to-one identification of valuations and absolute values: The valuation is assigned the absolute value which is defined as:

Conversely, the absolute value is assigned the valuation which is defined as: This will be used throughout the article.

Contents

Origin of the nameEdit

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 (see also the definition of the idele class group). The term "idele" is a variation of the term ideal. Both terms have a relation, see the theorem about the relation between the ideal class group and the idele class group. 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 that we want to have a look on all completions of   at once. A first glance, the Cartesian product could be a good candidate. However, the adele ring is defined with the restricted product (see next section). There are two reasons for this:

  • For each element of the global field   the valuations are zero for almost all places, which means 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 on the Cartesian product.

Definition of the adele ring of a global field Edit

Definition: the set of the finite adeles of a global field  Edit

The set of the finite adeles of a global field   named   is defined as the restricted product of   with respect to the   which means

 

This means, that the set of the finite adeles contains all   so that   for almost all   Addition and multiplication are defined component-wise. In this way   is a ring. The topology is the restricted product topology. That means that the topology is generated by the so-called restricted open rectangles, which have the following form:

 

where   is a finite subset of the set of all places of   containing   and   is open. In the following, we will use the term finite adele ring of   as a synonym for  

Definition: the adele ring of a global field  Edit

The adele ring of a global field   named   is defined as the product of the set of the finite adeles with the product of the completions at the infinite valuations. These are   or   their number is finite and they appear only in case, when   is an algebraic number field. That means

 

In case of a global function field, the finite adele ring equals the adele ring. We define addition and multiplication component-wise. As a result, the adele ring is a ring. The elements of the adele ring are called adeles of   In the following, we write

 

although this is generally not a restricted product.

Definition: the set of the  -adeles of a global field  Edit

Let   be a global field and   a subset of the set of places of   Define the set of the  -adeles of   as

 

If there are infinite valuations in   they are added as usual without any restricting conditions.

Furthermore, define

 

Thus,  

Example: the rational adele ring  Edit

We consider the case   Due to Ostrowski's theorem, we can identify the set of all places of   with   where we identify the prime number   with the equivalence class of the  -adic absolute value and we identify   with the equivalence class of the absolute value   on   defined as follows:

 

Next, we note, that the completion of   with respect to the places   is the field of the p-adic numbers   to which the valuation ring   belongs. For the place   the completion is   Thus, the finite adele ring of the rational numbers is

 

As a consequence, the rational adele ring is

 

We denote in short

 

for the adele ring of   with the convention  

Lemma: the difference between restricted and unrestricted product topologyEdit

The sequence in  

 

converges in the product topology with limit   however, it doesn't converge in the restricted product topology.

Proof: The convergence in the product topology corresponds to the convergence in each coordinate. The convergence in each coordinate is trivial, because the sequences become stationary. The sequence doesn't converge in the restricted product topology because for each adele   and for each restricted open rectangle   we have the result:   for   and therefore   for all   As a result, it stands, that   for almost all   In this consideration,   and   are finite subsets of the set of all places.

The adele ring does not have the subspace topology, because otherwise the adele ring would not be a locally compact group (see the theorem below).

Lemma: diagonal embedding of   in  Edit

Let   be a global field. There is a natural diagonal embedding of   into its adele ring  

 

This embedding is well-defined, because for each   it stands, that   for almost all   The embedding is injective, because the embedding of   in   is injective for each   As a consequence, we can view   as a subgroup of   In the following,   is a subring of its adele ring. The elements of   are the so-called principal adeles of  

Alternative definition of the adele ring of an algebraic number fieldEdit

Definition: profinite integersEdit

Define

 

that means   is the profinite completion of the rings   with the partial order  

With the Chinese Remainder Theorem, it can be shown that the profinite integers are isomorphic to the product of the integer p-adic numbers. It stands:

 

Lemma: alternative definition of the adele ring of an algebraic number fieldEdit

Define the ring

 

With the help of this ring the adele ring of the rational numbers can be written as:

 

This is an algebraic isomorphism. For an algebraic number field   it stands:

 

where we install on the right hand side the following topology: It stands, that   where the right hand side has   summands. We give the right hand side the product topology of   and transport this topology via the isomorphism onto  

Proof: We will first prove the equation about the rational adele ring. Thus, we have to show that   It stands   As a result, it is sufficient to show that   We will prove the universal property of the tensor product: Define a  -bilinear function   via   This function is obviously well-defined, because only a finite number of prime numbers divide the denominator of   It stands that   is  -bilinear.

Let   be another  -module together with a  -bilinear function   We have to show that there exists one and only one  -linear function   with the property:   We define the function   in the following way: For a given   there exists a   and a   such that   for all   Define   It can be shown that   is well-defined,  -linear and satisfies   Furthermore,   is unique with these properties. The general statement can be shown similarly and will be proved in the following section in general formulation.

The adele ring   in case of a field extension  Edit

Lemma: alternative description of the adele ring in case of  Edit

Let   be a global field. Let   be a finite field extension. In case K is an algebraic number field the extension is separable. If K is a global function field, it can be assumed as separable as well, see Weil (1967), p. 48f. In any case,   is a global field and thus   is defined. For a place   of   and a place   of   we define

 

if the absolute value   restricted to   is in the equivalence class of   We say the place   lies above the place   Define

 

Here   denotes a place of   and   denotes a place of   Furthermore, both products are finite.

Remark: We can embed   in   if   Therefore, we can embed   diagonal in   With this embedding the set   is a commutative algebra over   with degree  

It is valid, that

 

This can be shown with elementary properties of the restricted product.

The adeles of   can be canonically embedded in the adeles of   The adele   is assigned to the adele   with   for   Therefore,   can be seen as a subgroup of   An element   is in the subgroup   if   for   and if   for all   and   for the same place   of  

Lemma: the adele ring as a tensor productEdit

Let   be a global field and let   be a finite field extension. It stands:

 

This is an algebraic and topological isomorphism and we install the same topology on the tensor product as we defined it in the lemma about the alternative definition of the adele ring. In order to do this, we need the condition   With the help of this isomorphism, the inclusion   is given via the function

 

Furthermore, the principal adeles of   can be identified with a subgroup of the principal adeles of   via the map

 

Proof: Let   be a basis of   over   It stands, that

 

for almost all   see Cassels (1967), p. 61.

Furthermore, there are the following isomorphisms:

 

where   is the canonical embedding   and as usual   We take on both sides the restricted product with restriction condition  

 

Thus we arrive at the desired result. This proof can be found in Cassels (1967), p. 65.

Corollary: the adele ring of   as an additive group

Viewed as additive groups, the following is true:

 

where the left side has   summands. The set of principal adeles in   are identified with the set   where the left side has   summands and we consider   as a subset of  

Definition of the adele ring of a vector-space   over   and an algebra   over  Edit

Lemma: alternative description of the adele ringEdit

Let   be a global field. Let   be a finite subset of the set of all places of   which contains   As usual, we write   for the set of all infinite places of   Define

 

We define addition and multiplication component-wise and we install the product topology on this ring. Then   is a locally compact, topological ring. In other words, we can describe   as the set of all   where   for all   That means   for all  

Remark: Is   another subset of the set of places of   with the property   we note, that   is an open subring of  

Now, we are able to give an alternative characterisation of the adele ring. The adele ring is the union of all the sets   where   passes all the finite subsets of the whole set of places of   which contains   In other words:

 

That means, that   is the set of all   so that   for almost all   The topology of   is induced by the requirement, that all   become open subrings of   Thus,   is a locally compact, topological ring.

Let's fix a place   of   Let   be a finite subset of the set of all places of   containing   and   It stands:

 

Define

 

It stands:

 

Furthermore, define

 

where   runs through all finite sets fulfilling   Obviously it stands:

 

via the map   The entire procedure above can be performed also with a finite subset   instead of  

By construction of   there is a natural embedding of   in     Furthermore, there exists a natural projection  

Definition: the adele ring of a vector-space   over  Edit

The two following definitions are based on Weil (1967), p. 60ff. Let   be a global field. Let   be a  -dimensional vector-space over   where   We fix a basis   of   over   For each place   of   we write   and   We define the adele ring of   as

 

This definition is based on the alternative description of the adele ring as a tensor product. On the tensor product we install the same topology we defined in the lemma about the alternative definition of the adele ring. In order to do this, we need the condition   We install the restricted product topology on the adele ring  

We receive the result, that   We can embed   naturally in   via the function  

In the following, we give an alternative definition of the topology on the adele ring   The topology on   is given as the coarsest topology, for which all linear forms (linear functionals) on   that means linear maps   extending to linear functionals of   to   are continuous. We use the natural embedding of   into   respectively of   into   to extend the linear forms.

We can define the topology in a different way: Take a basis   of   over   This results in an isomorphism of   to   As a consequence the basis   induces an isomorphism of   to   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. This homeomorphism transfers the two topologies into each other.

In a formal way, it stands:

 

where the sums have   summands. In case of   the definition above is consistent with the results about the adele ring in case of a field extension  

Definition: the adele ring of an algebra   over  Edit

Let   be a global field and let   be a finite-dimensional algebra over   In particular,   is a finite-dimensional vector-space over   As a consequence,   is defined. We establish a multiplication on   based on the multiplication of  

It stands, that   Since, we have a multiplication on   and on   we can define a multiplication on   via

 

Alternatively, we fix a basis   of   over   To describe the multiplication of   it is sufficient to know, how we multiply two elements of the basis. There are   so that

 

With the help of the   we can define a multiplication on  

 

In addition to that, we can define a multiplication on   and therefore on  

As a consequence,   is an algebra with 1 over   Let   be a finite subset of   containing a basis of   over   We define   as the  -modul generated by   in   where   is a finite place of   For each finite subset   of the set of all places, containing   we define

 

It can be shown, that there is a finite set   so that   is an open subring of   if   contains   Furthermore, it stands, that   is the union of all these subrings. It can be shown, that in case of   the definition above is consistent with the definition of the adele ring.

Trace and norm on the adele ringEdit

Definition: trace and norm on the adele ringEdit

Let   be a finite extension of the global field   It stands   Furthermore, it stands   As a consequence, we can interpret   as a closed subring of   We write   for this embedding. Explicitly, it stands:   and this is true for all places   of   above   and for any  

Now, let   be a tower of global fields. It stands:

 

Furthermore, if we restrict the map   to the principal adeles,   becomes the natural injection  

Let   be a basis of the field extension   That means, that each   can be written as   where the   are unique. The map   is continuous. We define   depending on   via the equations

 

Now, we define the trace and norm of   as:

 

These are the trace and the determinant of the linear map     They are continuous maps on the adele ring.

Lemma: properties of trace and normEdit

Trace and norm fulfil the usual equations:

 

Furthermore, we note that for an   the trace   and the norm   are identical to the trace and norm of the field extension   For a tower of fields   it stands:

 

Moreover, it can be shown, that

 

Remark: The last two equations aren't obvious, see Weil (1967), p. 52ff respectively p. 64 or Cassels (1967), p. 74.

Properties of the adele ringEdit

In principle, to prove the following statements, we can reduce the situation to the case   or   The generalisation for global fields is often trivial.

Theorem: the adele ring is a locally compact, topological ringEdit

Let   be a global field. It stands, that   is a topological ring for every subset   of the set of all places. Furthermore,   is a locally compact group, that means, that the set   is locally compact and the group operation is continuous, that means that the map

 

is continuous and the map of the inverse is continuous, too, resulting in the continuous map

 

A neighbourhood system of   in   is a neighbourhood system of   in the adele ring. Alternatively, we can take all sets of the form   where   is a neighbourhood of   in   and   for almost all  

Idea of proof: The set   is locally compact, because all the   are compact and the adele ring is a restricted product. The continuity of the group operations can be shown with the continuity of the group operations in each component of the restricted product. A more detailed proof can be found in Deitmar (2010), p. 124, theorem 5.2.1.

Remark: The result above can be shown similarly for the adele ring of a vector-space   over   and an algebra   over  

Theorem: the global field is a discrete, cocompact subgroup in its adele ringEdit

The adele ring contains the global field as a discrete, cocompact subgroup. This means that   is discrete and   is compact in the topology of the quotient. In particular,   is closed in  

Proof: A proof can be found in Cassels (1967), p. 64, Theorem, or in Weil (1967), p. 64, Theorem 2. In the following, we reflect the proof for the case  

We have to show that   is discrete in   It is sufficient to show that there exists a neighbourhood of   which contains no more rational numbers. The general case follows via translation. Define

 

Then   is an open neighbourhood of   in   We have to show that   Let   be in   It follows that   and   for all   and therefore   Additionally, we have that   and therefore  

Next, we show that   is compact. Define the set

 

We show that each element in   has a representative in   In other words, we need to show that for each adele  , there exists   such that   Take an arbitrary   and let   be a prime number for which   Then there exists   with     and   We replace   by   This replacement changes the other places as follows:

Let   be another prime number. One has the following:   It follows that   (″ ″ 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 prime numbers for which the components of   aren't in   is reduced by 1. With an iteration, we arrive at the result that   exists with the property that   Now we select   such that   is in   Since   is in   it follows that   for   We consider the continuous projection   The projection is surjective. Therefore,   is the continuous image of a compact set, and thus compact by itself.

The last statement is a lemma about topological groups.

Corollary: Let   be a global field and let   be a finite-dimensional vector-space over   Then   is discrete and cocompact in  

Lemma: properties of the rational adele ringEdit

In a previous section, we defined   It stands

 

Furthermore, it stands, that   is unlimited divisible, which is equivalent to the statement, that the equation   has a solution   for each   and for each   This solution is generally not unique.

Furthermore, it stands, that   is dense in   This statement is a special case of the strong approximation theorem.

Proof: The first two equations can be proved in an elementary way. The next statement can be found in Neukirch (2007) on page 383. We will prove it. Let   and   be given. We need to show the existence of a   with the property:   It is sufficient to show this statement for   This is easily seen, because   is a field with characteristic unequal zero in each coordinate. In the following, we give a counter example, showing, that   isn't uniquely reversible. Let   and   be given. Then   fulfils the equation   In addition,   fulfils this equations as well, because   It stands, that   is well-defined, because there exists only a finite number of prime numbers, dividing   However, it stands, that   because by considering the last coordinate, we obtain  

Remark: In this case, the unique reversibility is equivalent to the torsion freedom, which is not provided here:   but   and  

We now prove the last statement. It stands:   as we can reach the finite number of denominators in the coordinates of the elements of   through an element   As a consequence, it is sufficient to show, that   is dense in   For this purpose, we have to show, that each open subset   of   contains an element of   Without loss of generality, we can assume

 

because   is a neighbourhood system of   in  

With the help of the Chinese Remainder Theorem, we can prove the existence of a   with the property:   because the powers of different prime numbers are coprime integers. Thus,   follows.

Definition: Haar measure on the adele ringEdit

Let   be a global field. We have seen that   is a locally compact group. Therefore, there exists a Haar measure   on   We can normalise   as follows: Let   be a simple function on   that means   where   measurable and   for almost all   The Haar measure   on   can be normalised so that for each simple, integrable function   the following product formula is satisfied:

 

where for each finite place, one has that   At the infinite places we choose Lebesgue measure.

We construct this measure by defining it on simple sets   where   is open and   for almost all   Since the simple sets generate the entire Borel  -algebra, the measure can be defined on the entire  -algebra. This can also be found in Deitmar (2010), p. 126, theorem 5.2.2.

Finitely, it can be shown that   has finite total measure under the quotient measure induced by the Haar measure on   The finiteness of this measure is a corollary of the theorem above, since compactness implies finite total measure.

Idele groupEdit

Definition of the idele group of a global field  Edit

Definition and lemma: topology on the group of units of a topological ringEdit

Let   be a topological ring. The group of units   together with the subspace topology, aren't a topological group in general. Therefore, we define a coarser topology on   which means that less sets are open. This is done in the following way: Let   be the inclusion map:

 

We define the topology on   as the topology induced by the subset topology on   That means, on   we consider the subset topology of the product topology. A set   is open in the new topology if and only if   is open in the subset topology. With this new topology   is a topological group and the inclusion map   is continuous. It is the coarsest topology, emerging from the topology on   that makes   a topological group.

Proof: We consider the topological ring   The inversion map isn't continuous. To demonstrate this, we consider the sequence

 

This sequence converges in the topology of   with the limit   The reason for this is, that for a given neighbourhood   of   it stands, that without loss of generality we can assume, that   is of form:

 

Furthermore, it stands, that   for all   Therefore, it stands, that   for all   big enough. The inversion of this sequence does not converge in the subset-topology of   We have shown this in the lemma about the difference between the restricted and the unrestricted product topology. In our new topology, the sequence and its inverse do not converge. This example shows that the two topologies are different in general. Now we show, that   is a topological group with the topology defined above. Since   is a topological ring, it is sufficient to show, that the function   is continuous. Let   be an open subset of   in our new topology, i.e.   is open. We have to show, that   is open or equivalently, that   is open. But this is the condition above.

Definition: the idele group of a global field  Edit

Let   be a global field. We define the idele group of   as the group of units of the adele ring of   which we write in the following as:

 

Furthermore, we define

 

We provide the idele group with the topology defined above. Thus, the idele group is a topological group. The elements of the idele group are called the ideles of  

Lemma: the idele group as a restricted productEdit

Let   be a global field. It stands

 

where these are identities of topological groups. The restricted product has the restricted product topology, which is generated by restricted open rectangles having the form

 

where   is a finite subset of the set of all places and   are open sets.

Proof: We will give a proof for the equation with  . The other two equations follow similarly. First we show that the two sets are equal:

 

Note, that in going from line 2 to 3,   as well as   have to be in   meaning   for almost all   and   for almost all   Therefore,   for almost all  

Now, we can show that 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   which is open in the topology of the idele group, meaning   is open, it stands that for each   there exists an open restricted rectangle, which is a subset of   and contains   Therefore,   is the union of all these restricted open rectangle and is therefore open in the restricted product topology.

Further definitions:

Define

 

and   as the group of units of   It stands

 

The idele group   in case  Edit

This section is based on the corresponding section about the adele ring.

Lemma: alternative description of the idele group in case  

Let   be a global field and let   be a finite field extension. It stands, that   is a global field and therefore the idele group   is defined. Define

 

Note, that both products are finite. It stands:

 

Lemma: embedding of   in  

There is a canonical embedding of the idele group of   in the idele group of   We assign an idele   the idele   with the property   for   Therefore,   can be seen as a subgroup of   An element   is in this subgroup if and only if his components satisfy the following properties:   for   and   for   and   for the same place   of  

The case of a vector-space   over   and an algebra   over  Edit

The following section is based on Weil (1967), p. 71ff.

Definition: