Adele ring

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In mathematics, the adele ring[a] (also adelic ring, ring of adeles or ring of adèles[1]) 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,[b] 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[edit]

Throughout this article, is a global field. That is, is a number field (a finite extension of ) or a global function field (a finite extension of ).

Let be a valuation of (we assume all valuations to be non-trivial). We write for the completion of with respect to If is discrete we write for the valuation ring of and for the maximal ideal of If this is a principal ideal we denote the uniformizing element by A non-Archimedean valuation is written as or and an Archimedean valuation as

By fixing a suitable constant there is a one-to-one identification of valuations and absolute values. The valuation is assigned the absolute value defined as:

Conversely, the absolute value is assigned the valuation defined as:

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

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" (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, i.e. for all places except a finite number. So, the global field can be embedded in the restricted product.
  • The restricted product is a locally compact space, while the Cartesian product is not. Therefore, we can't apply harmonic analysis to the Cartesian product.

The adele ring[edit]

Definition[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

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 a 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.

Lemma. There is a natural diagonal embedding of into its adele ring given by:

Proof. This embedding is well-defined, because for each we have 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

Let a 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,

The adele ring of rationals[edit]

By Ostrowski's theorem we can identify all places of with where we identify the prime number with the equivalence class of the -adic absolute value and with the equivalence class of the absolute value on defined as follows:

Next, we note, that the completion of with respect to the place 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

We will illustrate the difference between restricted and unrestricted product topology using a sequence in :

Lemma. Consider the following sequence in :
In product topology this sequences converges to 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).

Alternative definition for number fields[edit]

Definition (profinite integers). Set

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 p-adic integers:

Lemma. Define Then we have an algebraic isomorphism

Proof. We have:

As a result, it is sufficient to show that We will use the universal property of the tensor product. Define a -bilinear function

This function is well-defined because only finitely many primes divide the denominator of Let be another -module together with a -bilinear function We have to show that there exists one and only one -linear function such that We define as follows: for a given there exist and such that for all Define It can be shown that is well-defined, -linear and satisfies Furthermore, is unique with these properties.

With a similar proof we can show:

Lemma. For a number field

Remark. Using where there are summands, we give the right side the product topology and transport this topology via the isomorphism onto

The adele ring of a finite extension[edit]

If be a finite extension then is a global field and thus is defined. Let be a place of and a place of We say lies above denoted by if the absolute value restricted to is in the equivalence class of Define

Note that both products are finite. It follows from the elementary properties of the restricted product that:

Remark: If we can embed in Therefore, we can embed diagonally in With this embedding is a commutative algebra over with degree

can be canonically embedded in 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. If is a finite extension then both algebraically and topologically.

With the help of this isomorphism, the inclusion is given via the function

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

Proof.[2] Let be a basis of over Then for almost all

Furthermore, there are the following isomorphisms:

For the second we used the map:

in which is the canonical embedding and We take on both sides the restricted product with restriction condition

Corollary. As additive groups where the left side has summands.

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

The adele ring of vector-spaces and algebras[edit]

Lemma. Suppose is a finite set of places of and define
Define addition and multiplication component-wise and equip this ring with the product topology. Then is a locally compact, topological ring.

Remark. If is another finite set of places of containing then is an open subring of

Now, we are able to give an alternative characterization of the adele ring. The adele ring is the union of all sets :

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

Fix a place of Let be a finite set of places of containing and Define

Then:

Furthermore, define

where runs through all finite sets containing Then:

via the map The entire procedure above holds with a finite subset instead of

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

The adele ring of a vector-space[3][edit]

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 equip the adele ring with the restricted product topology. Then and we can embed in naturally via the map

We give an alternative definition of the topology on the adele ring Consider all linear maps: Using the natural embeddings and we extend these linear maps to: The topology on is the coarsest topology for which all these extensions are continuous.

We can define the topology in a different way. Take a basis of over This results in an isomorphism of Therefore the basis induces an isomorphism 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. More formally

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

The adele ring of an algebra[edit]

Let be a finite-dimensional algebra over In particular, is a finite-dimensional vector-space over As a consequence, is defined and Since we have a multiplication on and 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 a unit 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 set of places, containing we define

One can show there is a finite set so that is an open subring of if Furthermore is the union of all these subrings and that for the definition above is consistent with the definition of the adele ring.

Trace and norm on the adele ring[edit]

Let be a finite extension. Since and from Lemma above we can interpret as a closed subring of We write for this embedding. Explicitly for all places of above and for any

Let be a tower of global fields. Then:

Furthermore, restricted to the principal adeles is the natural injection

Let be a basis of the field extension Then each can be written as where 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 and they fulfil the usual equations:

Furthermore, for and are identical to the trace and norm of the field extension For a tower of fields we have:

Moreover, it can be proven that:[4]

Properties of the adele ring[edit]

Theorem. is a topological ring for every subset of set of all places. Furthermore, is a locally compact group.

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

Proof.[5] is locally compact, because are compact and the adele ring is a restricted product. The continuity of the group operations follows from the continuity of the group operations in each component of the restricted product.

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

Theorem. is discrete and is compact. In particular, is closed in

Proof.[6] We prove the case To show is discrete it is sufficient to show that there exists a neighbourhood of which contains no other rational number. The general case follows via translation. Define

is an open neighbourhood of We claim Let then and for all and therefore Additionally, we have and therefore

Next, we show that is compact. Define the set

We show that each element in has a representative in that is for each adele there exists such that Take an arbitrary and let be a prime for which Then there exists with and Replace with and let be another prime. Then:

Next we claim:

The reverse implication is trivially true. The implication is true, because the two terms of the strong triangle inequality are equal if the absolute values of both integers are different. As a consequence, the (finite) set of primes for which the components of are not in is reduced by 1. With iteration, we deduce that there exists such that Now we select such that Since it follows that for Consider the continuous projection Since it is surjective is the continuous image of a compact set, and thus compact.

The last statement is a lemma about topological groups.

Corollary. Let be a finite-dimensional vector-space over Then is discrete and cocompact in
Theorem. We have the following:
  • is a divisible group.[7]
  • is dense.

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

In order to show is divisible we need to show that for any and the equation has a solution It is sufficient to show that is divisible but this is true since is a field with positive characteristic in each coordinate.

For the last statement note that 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 is dense, that is each open subset contains an element of Without loss of generality, we can assume

because is a neighbourhood system of in Using the Chinese Remainder Theorem there exists such that Since powers of distinct primes are coprime follows.

Remark. is not uniquely divisible. Let and be given. Then and ( is well-defined, because only finitely many primes divide ), both satisfy the equation and clearly In this case, being uniquely divisible is equivalent to being torsion-free, which is not true for since but and

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

Haar measure on the adele ring[edit]

Since is a locally compact group, there exists a Haar measure on We can normalise as follows. Let be a simple function on that means where are 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.[8]

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.

The idele group[edit]

Definition[edit]

Definition. We define the idele group of as the group of units of the adele ring of that is The elements of the idele group are called the ideles of

Remark. We would like to equip with a topology so that it becomes a topological group. The subset topology inherited from is not a suitable candidate since the group of units of a topological ring equipped with subset topology may not be a topological group. For example the inverse map in is not continuous. The sequence

converges to To see this let be neighbourhood of without loss of generality we can assume:

Since for all for large enough. However as we saw above the inverse of this sequence does not converge in

Lemma. Let be a topological ring. Define:
Equipped with the topology induced from the product on topology on and 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. Since is a topological ring, it is sufficient to show that the inverse map is continuous. Let be open, then is open. We have to show is open or equivalently, that is open. But this is the condition above.

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

Lemma. Define Then the following identities of topological groups hold:
The restricted product has the restricted product topology, which is generated by restricted open rectangles of the form
where is a finite subset of the set of all places and are open sets.

Proof. We prove the identity for the other two follow similarly. First we show 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 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 rectangles and therefore is open in the restricted product topology.

Lemma. For each set of places, is a locally compact topological group.

Proof. The local compactness follows from the descriptions of the idele group as a restricted product. That is a topological group follows from the above discussion about the group of units of a topological ring.

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

where is a neighbourhood of and for almost all

Define

and let be its group of units. Then:

Since the idele group is a locally compact, 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

The idele group of a finite extension[edit]

Lemma. Let be a finite extension. Define
Note, that both products are finite. Then:
Lemma. There is a canonical embedding of in

Proof. 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 vector-spaces and algebras[edit]

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

Definition. Let be a finite-dimensional vector space over Define:

This is an algebra over and where the inverse of a linear map exists if and only if the determinant is non-zero. is an open subset of is a topological field, therefore is closed, since is continuous, is open.

The idele group of an algebra[edit]

Let be a finite-dimensional algebra over Since is not a topological group with the subset-topology in general, we equip with the topology similar to above and call the idele group. The elements of the idele group are called idele of

Proposition. Let be a finite subset of containing a basis of over For each finite place of let be the -module generated by in There exists a finite set of places containing such that for all is a compact subring of Furthermore, contains For each is an open subset of and the map is continuous on As a consequence, the function maps homeomorphically on its image in For each the are the elements of mapping in with the function above. Therefore, is an open and compact subgroup of [9]

Alternative characterisation of the idele group[edit]

Proposition. Let be a finite set of places. Then
is an open subgroup of where is the union of all [10]
Corollary. In the special case when for each finite set of places
is an open subgroup of Furthermore, is the union of all

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 Then and therefore, we have in injective group homomorphism

Since is in in particular is invertible, is invertible too, because Therefore 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.

The Idele class group[edit]

The group of units of can be embedded diagonally in the idele group

Since is a subset of for all the embedding is well-defined and injective. Furthermore, is discrete and closed in

Corollary. is a discrete subgroup of

Defenition. In analogy to the ideal class group, the elements of in are called principal ideles of The quotient group is called idele class group of This group is related to the ideal class group and is a central object in class field theory.

Remark. is closed in therefore is a locally compact topological group and a Hausdorff space.

Let be a finite extension. 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 [11]

Properties of the idele group[edit]

Absolute value on and -idele[edit]

For a given idele we define its absolute value as:

Since this product is finite and therefore well-defined. This definition can be extended to the whole adele ring by allowing infinite products. This means, we consider convergence in These infinite products vanish so that the absolute value function vanishes on In the following, we use to denote this function on respectively

Theorem. is a continuous group homomorphism.

Proof. Let

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 -idele as:

is a subgroup of Since it is a closed subset of Finally the -topology on equals the subset-topology of on [12][13]

Artin's Product Formula. for all

Proof.[14] We prove the formula for number fields, the case of a global function field can be proved similarly. Let be a number field and We have to show:

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 We have:

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

Lemma. There exists a constant depending only on such that for every satisfying there exists a such that for all [15]
Corollary. 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 from the lemma. Let be a uniformizing element of Define the adele via with minimal, so that for all Then for almost all Define with so that This works, because for almost all Using the lemma there exists a so that for all

Theorem. is discrete and cocompact in

Proof.[16] Since is discrete in it is also discrete in To prove is compact it is sufficient to show 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 We show there exists so that It stands, that

and therefore

It follows that

By the lemma, there exists an such that for all and therefore

Theorem. There is a canonical isomorphism Furthermore, is a set of representatives of in other words:
Additionally, the absolute value function induces the following isomorphisms of topological groups:
Consequently, is a set of representatives of [17]

Proof. Consider the map

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 The absolute value of this element is and therefore It follows, that It stands, that