Adele ring

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In mathematics the adele ring 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.

Notation: During the whole article, is a global field. That means, that 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 write for a place of that means is a representative of an equivalence class of valuations. 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 name[edit]

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 is an invention of the French mathematician Claude Chevalley (1909-1984) and stands for ″ideal element″ (abbreviated: id.el.). The term adele 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, the Cartesian product 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 concerning 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 topology[edit]

The sequence in

converges in the product topology with limit however, it doesn't converges 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 convergence 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 field[edit]

Definition: profinite integers[edit]

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 field[edit]

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 an given there exists a and a such that for all Define It can be shown, that is well-defined, -linear and satisfies Furthermre, 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 on is in the equivalence class of We say, the place lies above the place Define

Respect, that 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 product[edit]

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 ring[edit]

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 ring[edit]

Definition: trace and norm on the adele ring[edit]

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 norm[edit]

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 ring[edit]

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 ring[edit]

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 ring[edit]

The adele ring contains the global field as a discrete, cocompact subgroup. That 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. Via translation, we can show the general case. Define

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

Now, we show, that is compact. Define the set

We show, that, each element in has a representative in This means, we have to show, that for each adele there exists a so that Take an arbitrary Let be a prime number, for which There exists a with and We replace by This replacement change the others places as follows:

Let be another prime number. It stands: 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 so 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 It stands, that is discrete and cocompact in

Lemma: properties of the rational adele ring[edit]

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 ring[edit]

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 stands the product formula:

where for each finite place, it stands At the infinite places we choose the 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 set, the measure can be defines on the entire borel set. This can also be found in Deitmar (2010), p. 126, theorem 5.2.2.

Finitely, it can be shown, that has finite measure in the quotient measure, which is induced by the Haar measure on The finite measure is a corollary of the theorem above, because compactness implies finite measure.

Idele group[edit]

Definition of the idele group of a global field [edit]

Definition and lemma: topology on the group of units of a topological ring[edit]

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 an 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 don 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 product[edit]

Let be a global field. It stands

where these are identities of topological rings. 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 sets 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:

Let be a finite-dimensional vector-space over where is a global field. Define:

This is an algebra over It stands, that where for a linear map the inverse function exists if and only if the determinant is not equal to Since is a global field, which in particular means that is a topological field, is an open subset of The reason for this is, that Since is closed and the determinant is continuous, is open.

Definition and proposition: the idele group of an algebra over [edit]

Let be a finite-dimensional algebra over where is global field. We consider the group of units of The map is generally not continuous with the subset-topology. Therefore, the group of units is not a topological group in general. On we install the topology we defined in the section about the group of units of a topological ring. With this topology, we call the group of units of the idele group The elements of the idele group are called idele of

Let be a finite subset of containing a basis of over For each finite place of we call the -modul generated by in As before, there exists a finite subset of the set of all places, containing so that it stands for all that is a compact subring of Furthermore, contains the group of units of In addition to that, it stands, that is an open subset of for each and that the map is continuous on As a consequence, the function maps homeomorphic on the image of this map in For each it stands, that the are the elements of mapping in with the function above. Therefore, is an open and compact subgroup of A proof of this statement can be found in Weil (1967), p. 71ff.

Proposition: alternative characterisation of the idele group[edit]

We consider the same situation as before. Let be a finite subset of the set of all places containing It stands, that

is an open subgroup of where is the union of all the A proof of this statement can be found in Weil (1967), p. 72.

Corollary: the case

We consider the case For each finite subset of the set of all places of containing it stands, that the group

is an open subgroup of Furthermore, it stands, that is the union of all these subgroups

Norm on the idele group[edit]

We want to transfer the trace and the norm from the adele ring to the idele group. It turns out, that 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 be in It stands, that and therefore, we have in injective group homomorphism

Since is in in particular is invertible, is invertible too, because Therefore, it stands, that As a consequence, the restriction of the norm-function introduces the following function:

This function is continuous and fulfils the properties of the lemma about the properties from the trace and the norm.

Properties of the idele group[edit]

Lemma: is a discrete subgroup of [edit]

The group of units of the global field can be embedded diagonal in the idele group

Since is a subset of for all the embedding is well-defined and injective.

Furthermore, it stands, that is discrete and closed in This statement can be proved with the same methods like the corresponding statement about the adele ring.

Corollary is a discrete subgroup of

Definition: idele class group[edit]

In number theory, we can define the ideal class group for a given algebraic number field. In analogy to the ideal class group, we call the elements of in principal ideles of The quotient group is the so-called idele class group of This group is related to the ideal class group and is a central object in class field theory.

Remark: Since is closed in it follows, that is a locally compact, topological group and a Hausdorff space.

Let be a finite field extension of global fields. The embedding induces an injective map on the idele class groups:

This function is well-defined, because the injection obviously maps onto a subgroup of The injectivity is shown in Neukirch (2007), p. 388.

Theorem: the idele group is a locally compact, topological group[edit]

For each subset of the set of all places, is a locally compact, topological group.

Remark: In general, equipped with the subset topology is not a topological group, because the inversion function isn't continuous.

The local compactness follows from the descriptions of the idele group as a restricted product. The fact, that the idele group is a topological group follows from considerations about the group of units of a topological ring.

Since the idele group is a locally compact group, there exists a Haar measure on it. This can be normalised, so that This is the normalisation used for the finite places. In this equations, 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

A neighbourhood system of in is a neighbourhood system of in Alternatively, we can take all sets of the form:

where is an neighbourhood of in and for almost all

Definition: absolute value on and the set of the -idele of [edit]

Let be a global field. We define an absolute value function on the idele group: For a given idele we define:

Since this product is finite and therefore well-defined. This definition can be extended onto the whole adele ring by allowing infinite products. This means, we consider convergence in These infinite products are so that the absolute value function is zero on In the following, we will write for this function on respectively

It stands, that the absolute value function is a continuous group homomorphism, which means that the map is a continuous group homomorphism.

Proof: Let and be in It stands:

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 the question, whether the absolute value function is continuous on the local fields However, this is clear, because of the reverse triangle inequality.

We define the set of the -idele, as the following:

It stands, that is a subgroup of In literature, the term is used as a synonym for the set of the -Idele. We will use in the following.

It stands, that is a closed subset of because

The -topology on equals the subset-topology of on This statement can be found in Cassels (1967), p. 69f.

Theorem: Artin's product formula[edit]

Let be a global field. The homomorphism of to fulfils: In other words, it stands, that for all Artin's product formula says, that is a subset of

Proof: There are many proofs for the product formula. The one shown in the following is based on Neukirch (2007), p. 195. In the case of an algebraic number field, the main idea is to reduce the problem to the case The case of a global function field can be proved similarly.

Let be in We have to show, that

It stands, that and therefore for each for which the corresponding prime ideal does not divide the principal ideal This is valid for almost all

It stands:

Note that in going from line 1 to line 2, we used the identity where is a place of and is a place of lying above Going from line 2 to line 3, we use a property from the norm. We note, that the norm is in Therefore, without loss of generality, we can assume that Then possesses a unique integer factorisation:

where