Adele ring

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: 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.

Origin of the name

In local class field theory, the group of units of the local field plays a central role. In global class field theory, the idele class group takes this role (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" (idèle) 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:

Definition of the adele ring of a global field

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

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

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

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

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

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 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

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

Definition: profinite integers

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

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

Lemma: alternative description of the adele ring in case of

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

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

Lemma: alternative description of the adele ring

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

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

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

Definition: trace and norm on the adele ring

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

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

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

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

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

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

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

Definition of the idele group of a global field

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

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

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

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

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

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

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

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

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

Lemma: is a discrete subgroup of

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

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

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

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

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 is for almost all Due to Ostrowski's theorem every absolute values on is equivalent to either the usual real absolute value or a -adic absolute value, we can conclude, that

which is the desired result. In the mathematical literature many more proofs of the product formula can be found.

Theorem: characterisation of

Let be a -dimensional vector-space over Define Furthermore, let be in We obtain the following equivalent statements:

If one of the three points above is true, we can conclude that Moreover, it stands, that the maps and are homomorphism of to respectively to A proof of this statement can be found in Weil (1967), p. 73f.

Corollary: Let be a finite-dimensional algebra over und let be in We obtain the following equivalent statements:

If one of the three points above is true, we can conclude that Moreover, it stands, that the maps and are homomorphism of to respectively to Based on this statement an alternative proof of the product formula is possible, see Weil (1967), p. 75.

Theorem: is a discrete and cocompact subgroup in the set of the -idele

Prior to formulate the theorem, we require the following lemma:

Lemma: Let be a global field. There exists a constant depending only on the global field such that for every with the property there exists a such that for all

A proof of this lemma can be found in Cassels (1967), p. 66 Lemma.

Corollary: Let be a global field. Let be a place of and let be given for all with the property for almost all Then, there exists a so that for all

Proof: Let be the constant form of the prior lemma. Let be a uniformizing element of Define the adele via with minimal, so that for all It stands, that for almost all Define with so that This works, because for almost all Using the lemma above, there exists a so that for all

Now we are ready to formulate the theorem.

Theorem: Let be a global field, then is discrete in and the quotient is compact.

Proof: The fact that is discrete in implies that is also discrete in

We have to show, that is compact. This proof can be found in Weil (1967), p. 76 or in Cassels (1967), p. 70. In the following, we will outline Cassels' (1967) idea of proof:

It is sufficient to show, that there exists a compact set such that the natural projection is surjective, because is continuous. Let with the property be given, where is the constant of the lemma above. Define

Obviously, is compact. Let be in We show, that there exists an so that It stands, that

and therefore

It follows, that

Because of the lemma, there exists an such that for all and therefore This ends the proof of the theorem.

Theorem: some isomorphisms in case

In case there is a canonical isomorphism Furthermore, is a set of representatives of that means, that Additionally, the absolute value function induces the following isomorphisms of topological groups:

Consequently, is a set of representatives of This theorem is part of theorem 5.3.3 on page 128 in Deitmar (2010).

Proof: Consider the map via This map is well-defined, since for all and therefore Obviously, this map is a continuous, group homomorphism. To show injectivity, let As a result, it exists a so that By considering the infinite place, we receive and therefore To show the surjectivity, let be in The absolute value of this element is and therefore It follows, that It stands, that and therefore the map is surjective, since für alle The other isomorphisms are given by: via and via

Theorem: relation between ideal class group and idele class group

For an algebraic number field we define It stands:

Here, is the group of fractional ideals of and is the ideal class group of the Dedekind domain It stands, that is the ring of integers of the algebraic number field Per definition, it stands, that

Proof: In the following, we will use the fact, that for an algebraic number field there is a one-to-one correspondence between the finite places of and the prime ideals of which are different from

Let be a finite place of and let be a representative of the equivalence class Define

Then is is a prime ideal in The map is a bijection between the set of the finite places of and the set of all prime ideals of The inversion map is given by the following:

A given prime ideal is mapped to the valuation given by

Now, we can prove the theorem. The following map is well-defined:

where is the corresponding prime ideal to the place The map is obviously a surjective homomorphism. It stands, that The first isomorphism of the theorem follows now using the fundamental theorem on homomorphism.

Now, we divide both sides from our map by This is possible, because

Please, note the abuse of notation: On the left hand side in line 1 of this chain of equations, stands for the map defined above. Later, we use the embedding of into In line 2, we use the definition of the map. Finally, we use the fact, that the ring of integers is a Dedekind domain and therefore each ideal, in particular the principal ideal can be written as a product of prime ideals. In other words, the map is a -equivariant group homomorphism. As a consequence, the map above induces a surjective homomorphism

To prove the second isomorphism, we have to show, that Consider be in Then because for all On the other hand, consider with which allows to write As a consequence, there exists a representative, such that: Consequently, and therefore We have proved the second isomorphism of the theorem.

For the last isomorphism of the theorem, we note, that the map induces a surjective group homomorphism

with This ends the proof of the theorem.

Remark: The map is continuous, if we install the following topology on the sets which are considered: On we install the idele topology. On we install the discrete topology. Continuity follows, since we can show, that is open for each It stands, that is open, where so that

Theorem: decomposition of and

Let be a global field. If the characteristic of is greater than zero, it stands, that If the characteristic of equals then where is a closed subgroup of which is isomorph to Furthermore, it stands, that

where if or if

Proof: Let be the characteristic of For each place of stands, that the characteristic of equals so that for each the element is in the subgroup of generated by It follows, that for each the number is in the subgroup of generated by It follows, that the image of the homomorphism is a discrete subgroup of generated by Since this group isn't trivial, meaning it is generated by a for a Choose so that it follows, that is the direct product of and the subgroup generated by This subgroup is discrete and isomorphic to

If the characteristic of equals we write for idele fulfilling for all finite places and for all infinite places of where It stands, that the map is an isomorphism of in a closed subgroup of and it stands, that The isomorphism is given by multiplication:

Obviously, is a homomorphism. To show injectivity, let Since for it stands that for Moreover, it exists a so that for Therefore, for In addition to that, since it follows, that where is the number of infinite places of As a consequence, it stands, that und therefore is injectiv. To show surjectivity, let be in We define and furthermore, we define for and for Define It stands, that Therefore, is surjective.

The other equations follow similarly.

Theorem: characterisation of the idele group

Let be an algebraic number field. There exists a finite subset of the set of all places, such that

Proof: In this proof, we will use the fact, that the class number of an algebraic number field is finite. Let be the ideals, representing the classes in These ideals are generated of a finite number of prime ideals Let be a finite set of places, which includes the infinite places of and those finite places corresponding to the prime ideals

We consider the isomorphism

which is induced by

In the following, we prove the statement for the finite places, because at the infinite places the statement is obvious. The inclusion ″″ is obvious. Let The corresponding ideal belongs to a class meaning for a principal ideal The idele maps to the ideal under the map That means Since the prime ideals in are in it follows for all that means for all It follows, that therefore The general proof of this theorem for any global field is given in Weil (1967), p. 77.

Applications

Finiteness of the class number of an algebraic number field

In this section, we want the show that the class number of an algebraic number field is finite. Of course, there are many different proofs of this statement. In the proof of the characterisation of the idele group, we already used this fact.

Theorem: (finiteness of the class number of an algebraic number field) Let be an algebraic number field. It stands, that

Proof: The map is surjective and therefore is the continuous image of the compact set Thus, is compact. In addition, is discrete, thus it is finite.

Remark: There is a similar result for the case of a global function field. In this case, the so-called divisor group is defined. It can be shown, that the quotient of the set of all divisors of degree by the set of the principal divisors is a finite group. For more information, see Cassels (1967), p. 71.

Group of units and Dirichlet's unit theorem

Some definitions

Let be a global field. Let be a finite subset of the set of all places, containing Define

It is valid, that is a subgroup of containing all elements which fulfil for all Since is discrete in it follows, that is a discrete subgroup of and with the same argument, is discrete in

An alternative definition of is, that where is a subring of defined by As a consequence, contains all elements which fulfil for all

Let It stands, that the set is finite. In order to prove this statement, we define

It stands, that is compact and the set described above is the intersection of with the discrete subgroup in The finiteness follows from these arguments.

Define where the second equal sign is true because of Artin's product formula. Define It stands

for each finite subset of the set of all places of containing

Theorem: roots of unity of

It stands, that is a finite, cyclic group, containing all roots of unity of Furthermore, it is valid, that is the group of all roots of unity of

Proof: It stands, that The last set is compact. Furthermore, is discrete in thus is finite, because is a subset of a compact set and is discrete. Because of Artin's product formula, it stands for all that for all It follows, that is a finite subgroup of Since is a field, is cyclic. It is obvious that each root of unity of is in since roots of unity of all have absolute value and therefore have valuation Suppose, that there exists a which isn’t a root of unity of It follows, that for all This contradicts the finiteness of the group

Theorem: generalised form of Dirichlet's unit theorem

Let the situation be as above. It stands that is the direct product of the group and a group being isomorphic to We note, that if and that if A proof can be found in Weil (1967), p. 78f. or in Cassels (1967), p. 72f.

Theorem: Dirichlet's unit theorem

Let be an algebraic number field. It stands

where is the finite, cyclic group of all roots of unity of and is the number of real embeddings of and is the number of conjugate pairs of complex embeddings of It stands, that

Remark: The theorem above is a generalisation of Dirichlet's unit theorem. For an algebraic number field define and receive the Dirichlet's unit theorem. In literature, this theorem is also called "Theorem of the units″. Of course, Dirichlet's unit theorem is older than the theorems given above and can be proved on its own. With the help of the Dirichlet's unit theorem, we can prove the compactness of in an alternative way.

Proof of this remark:

We already know, that

Furthermore, it stands, that

In addition to that, it stands that

Approximation theorems

Theorem: weak approximation theorem

Let be inequivalent, non-trivial valuations of the field Let In particular, these are topological spaces. We embed diagonal in It stands, that is everywhere dense in In other words, for each and for each there exists a such that

A proof can be found in Cassels (1967), p. 48f.

Theorem: strong approximation theorem

Let be a global field. Let be a place of Define

Then is dense in A proof can be found in Cassels (1967), p. 67f.

Remark: The global field is discrete in its adele ring. To obtain this result, we had to include all places of the global field. The strong approximation theorem tells us that, if we omit one place (or more), the property of discreteness of is turned into a denseness of

″Local–global″ and Hasse principle

Local and global:

Let be a finite extension of the global field We define as the global extension. Let be a place of and let be a place of lying above We define the (finite) extension as the local extension. Where do these names come from? In order to understand this, we consider the case of a global function field, for example although this isn't a global field. Let be a finite extension. The elements of are algebraic functions on a Riemann surface, a global object. On the other hand, if we consider the extension we change from studying these functions globally to a local one, which is equivalent to consider their power series. For more information, see Neukirch (2007), p. 169.

Theorem: Minkowski–Hasse

A quadratic form on the global field is zero, if and only if, the quadratic form is zero in each completion

Remark: This is the Hasse principle for quadratic forms. For polynomials of degree larger than 2 the Hasse principle isn't valid in general.

Remark: The idea of the local–global principle is to solve a given problem of an algebraic number field by doing so in its completions and then concluding on a solution in

Characters on the adele ring

Definition: character group

Let be a locally compact, abelian group. Define the character group of as the set of all characters of that means the set of all continuous group homomorphism of to We give the topology of the uniform convergence on compact subgroups of It can be shown, that is also a locally compact, abelian group.

Theorem: the adele ring is self-dual

Let be a global field. The adele ring is self-dual, that means, that

Proof: In a first step, we show that each is self-dual by fixing one character. We exemplify this for the case by defining via Now we consider the map with or in other words, It can be shown that is an isomorphism which respects topologies. In a second step the problem for the adele ring is treated by reducing it to a problem in the local coordinates.

Theorem: Algebraic and continuous dual space of the adele ring

Let be a global field and let be a non-trivial character of which is trivial on Let be a finite-dimensional vector-space over Let be its algebraic dual space and let be the algebraic dual space of Furthermore, let be the topological dual of Then the formula for all determines an isomorphism of onto where and On this occasion and are the bilinear pairings on and on Moreover, if fulfils for all then it stands, that A proof can be found in Weil (1967), p. 66.

With the help of the characters of we can do Fourier analysis on the adele ring (for more see Deitmar (2010), p. 129ff).

Tate's thesis

John Tate determines in his thesis "Fourier analysis in number fields and Heckes Zetafunctions" (see Cassels (1967)) results about Dirichlet L-functions by using Fourier analysis on the adele ring and the idele group. Therefore, the adele ring and the idele group have been applied to study the Riemann zeta function and more general to study the zeta functions and the L-functions. We can define adelic forms of these functions and we can represent these functions as integrals over the adele ring, respectively the idele group, which respect to the corresponding Haar measures. Out of this, we can show functional equations and meromorphic continuations of these functions. For illustration purposes, we provide an example. For each complex number with it stands

where is the normalised Haar measure on with which is extended by zero on the finite adele ring. Note that it is unique. The equation above shows, that we can write the Riemann zeta function as an integral over (a subset of) the adele ring. A proof can be found in Deitmar (2010), p. 128, Theorem 5.3.4. See also p. 139ff for more information on Tate's thesis.

Automorphic forms

We consider the case

In newer mathematical approaches, automorphic forms are described as being a function on the group satisfying several additional conditions. For this purpose, we define and as the centre of the group It stands, that We define an automorphic form as an element of the vector-space For studying automorphic forms, it is important to know the representations of the group which are described in the tensor product theorem. It is also possible to study automorphic L-function, which can be described as an integral over the group Further information can be found in Deitmar (2010) in the chapter about the automorphic representations of the adele group and in the chapter about the automorphic L-functions.

Further applications

A generalisation of Artin reciprocity law leads to the connection of representations of and of Galois representations of (Langlands program).

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

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

Notes

  1. Also spelled: adèle /əˈdɛl/.
  2. Also spelled: idèle /ɪˈdɛl/.

Literature

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