Adele ring


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, 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.

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. The term "idele" is an invention of the French mathematician Claude Chevalley and stands for "ideal element". The term "adele" stands for additive idele.
The idea of the adele ring is to look at all completions of at once. At first glance, the Cartesian product could be a good candidate. However, the adele ring is defined with the restricted product. There are two reasons for this:

Global fields

Throughout this article, is a global field, meaning it is either a number field or a global function field. By definition a finite extension of a global field is itself a global field.

Valuations

For a valuation of 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 We assume all valuations to be non-trivial.
There is a one-to-one identification of valuations and absolute values. Fix a constant 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 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
Define and let be its group of units. Then

Finite extensions

Let be a finite extension of the global field 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.
If we can embed in Therefore, we can embed diagonally in With this embedding is a commutative algebra over with degree

The adele ring

The set of finite adeles of a global field denoted is defined as the restricted product of with respect to the
It is equipped with the restricted product topology, the topology generated by restricted open rectangles, which have the following form:
where is a finite set of places and are open. With component-wise addition and multiplication is also a ring.
The adele ring of a global field is defined as the product of with the product of the completions of at its infinite places. The number of infinite places is finite and the completions are either or In short:
With addition and multiplication defined as component-wise 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.
Remark. Global function fields do not have any infinite places and therefore the finite adele ring equals the adele ring.
Proof. If then for almost all This shows the map is well-defined. It is also injective because the embedding of in is injective for all
Remark. By identifying with its image under the diagonal map we regard it as a subring of The elements of are called the principal adeles of
Definition. Let be a set of places of Define the set of the -adeles of as
Furthermore if we define
we have:

The adele ring of rationals

By Ostrowski's theorem the places of are where we identify a prime with the equivalence class of the -adic absolute value and with the equivalence class of the absolute value defined as:
The completion of with respect to the place is with valuation ring For the place the completion is Thus:
Or for short
We will illustrate the difference between restricted and unrestricted product topology using a sequence in :
Proof. In product topology convergence corresponds to the convergence in each coordinate, which is trivial because the sequences become stationary. The sequence doesn't converge in restricted product topology, for each adele and for each restricted open rectangle we have: for and therefore for all As a result for almost all In this consideration, and are finite subsets of the set of all places.

Alternative definition for number fields

Definition. We define profinite integers as the profinite completion of the rings with the partial order i.e.,
Proof. This follows from Chinese Remainder Theorem.
Proof. We will use the universal property of the tensor product. Define a -bilinear function
This is well-defined because for a given with co-prime there are only finitely many primes dividing Let be another -module with a -bilinear map We have to show factors through uniquely, i.e., there exists a unique -linear map such that We define as follows: for a given there exist and such that for all Define One can show is well-defined, -linear, satisfies and is unique with these properties.
Proof.
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

If be a finite extension then is a global field and thus is defined and We claim can be identified with a subgroup of Map to where for Then is in the subgroup if for and for all lying above the same place of
With the help of this isomorphism, the inclusion is given by
Furthermore, the principal adeles in can be identified with a subgroup of principal adeles in via the map
Proof. 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 respect to
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

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-spaceThe definitions are based on

Let be a finite dimensional vector-space over and a basis for over For each place of we write:
We define the adele ring of as
This definition is based on the alternative description of the adele ring as a tensor product equipped with the same topology we defined when giving an alternate definition of adele ring for number fields. We equip with the restricted product topology. Then and we can embed in naturally via the map
We give an alternative definition of the topology on Consider all linear maps: Using the natural embeddings and 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. Fixing a basis for over results in an isomorphism Therefore fixing a 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 which 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

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:
As a consequence, is an algebra with a unit over Let be a finite subset of containing a basis for over For any finite place we define as the -module generated by in For each finite set of places, 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 for the definition above is consistent with the definition of the adele ring.

Trace and norm on the adele ring

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:

Properties of the adele ring

Remark. The result above also holds for the adele ring of vector-spaces and algebras over
Proof. We prove the case To show is discrete it is sufficient to show the existence of 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 compactness, define:
We show each element in has a representative in that is for each there exists such that Let be arbitrary and 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 set of primes for which the components of are not in is reduced by 1. With iteration, we deduce there exists such that Now we select such that Then The continuous projection is surjective, therefore as the continuous image of a compact set, is compact.
Proof. The first two equations can be proved in an elementary way.
By definition is divisible if for any and the equation has a solution It is sufficient to show 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 By 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
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

Definition. A function is called simple if where are measurable and for almost all

The idele group

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
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.
Definition. For a subset of places of set:
Proof. We prove the identity for the other two follow similarly. First we show the two sets are equal:
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, so 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.
Proof. The local compactness follows from the description of as a restricted product. It being a topological group follows from the above discussion on 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
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

Proof. We map to 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 algebrasThis section is based on

The idele group of an algebra

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

Alternative characterisation of the idele group

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 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 it is invertible, is invertible too, because Therefore As a consequence, the restriction of the norm-function introduces a continuous function:

The Idele class group

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

Properties of the idele group

Absolute value on I_K and 1-idele

Definition. For define: Since is an idele this product is finite and therefore well-defined.
Remark. The definition can be extended to by allowing infinite products. However these infinite products vanish and so vanishes on We will use to denote both the function on and
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 whether is continuous on However, this is clear, because of the reverse triangle inequality.
Definition. 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
Proof. We prove the formula for number fields, the case of global function fields can be proved similarly. Let be a number field and We have to show:
For a finite place for which the corresponding prime ideal does not divide we have and therefore This is valid for almost all We have:
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 of the norm. We note the norm is in so without loss of generality we can assume Then possesses a unique integer factorisation:
where is for almost all By Ostrowski's theorem all absolute values on are equivalent to the real absolute value or a -adic absolute value. Therefore:
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 By the Lemma there exists so that for all
Proof. Since is discrete in it is also discrete in To prove the compactness of let is the constant of the Lemma and suppose satisfying is given. Define:
Clearly is compact. We claim the natural projection is surjective. Let be arbitrary, then:
and therefore
It follows that
By the Lemma there exists such that for all and therefore proving the surjectivity of the natural projection. Since it is also continuous the compactness follows.
Proof. Consider the map
This map is well-defined, since for all and therefore Obviously is a continuous group homomorphism. Now suppose Then there exists such that By considering the infinite place we see proving injectivity. To show surjectivity let The absolute value of this element is and therefore
Hence and we have:
Since
we conclude is surjective.
Proof. The isomorphisms are given by:

Relation between ideal class group and idele class group

Proof. Let be a finite place of and let be a representative of the equivalence class Define
Then is a prime ideal in The map is a bijection between finite places of and non-zero prime ideals of The inverse is given as follows: a prime ideal is mapped to the valuation given by
The following map is well-defined:
The map is obviously a surjective homomorphism and The first isomorphism follows from fundamental theorem on homomorphism. Now, we divide both sides 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
that is a Dedekind domain and therefore each 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 Consider 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 note that induces a surjective group homomorphism with
Remark. Consider with the idele topology and equip with the discrete topology. Since is open for each is continuous. It stands, that is open, where so that

Decomposition of I_K and C_K

Proof. For each place of so that for all belongs to the subgroup of generated by Therefore for each is in the subgroup of generated by Therefore the image of the homomorphism is a discrete subgroup of generated by Since this group is non-trivial, it is generated by for some Choose so that then is the direct product of and the subgroup generated by This subgroup is discrete and isomorphic to
For define:
The map is an isomorphism of in a closed subgroup of and The isomorphism is given by multiplication:
Obviously, is a homomorphism. To show it is injective, let Since for it stands that for Moreover, it exists a so that for Therefore, for Moreover implies where is the number of infinite places of As a consequence and therefore is injective. To show surjectivity, let We define and furthermore, we define for and for Define It stands, that Therefore, is surjective.
The other equations follow similarly.

Characterisation of the idele group

Proof. The class number of a number field is finite so let be the ideals, representing the classes in These ideals are generated by a finite number of prime ideals Let be a finite set of places containing and the finite places corresponding to Consider the isomorphism:
induced by
At infinite places the statement is obvious so we prove the statement for finite places. 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

Applications

Finiteness of the class number of a number field

In the previous section we used the fact that the class number of a number field is finite. Here we would like to prove this statement:
Proof. The map
is surjective and therefore is the continuous image of the compact set Thus, is compact. In addition it is discrete and so 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.

Group of units and Dirichlet's unit theorem

Let be a finite set of places. Define
Then is a subgroup of containing all elements satisfying for all Since is discrete in is a discrete subgroup of and with the same argument, is discrete in
An alternative definition is: where is a subring of defined by
As a consequence, contains all elements which fulfil for all
Proof. Define
is compact and the set described above is the intersection of with the discrete subgroup in and therefore finite.
Proof. All roots of unity of have absolute value so For converse note that Lemma 1 with and any implies is finite. Moreover for each finite set of places Finally Suppose there exists which is not a root of unity of Then for all contradicting the finiteness of
Remark. The Unit Theorem is a generalisation of Dirichlet's Unit Theorem. To see this let be a number field. We already know that set and note Then we have:

Approximation theorems

Remark. The global field is discrete in its adele ring. The strong approximation theorem tells us that, if we omit one place, the property of discreteness of is turned into a denseness of

Hasse principle

Remark. This is the Hasse principle for quadratic forms. For polynomials of degree larger than 2 the Hasse principle isn't valid in general. The idea of the Hasse principle is to solve a given problem of a number field by doing so in its completions and then concluding on a solution in

Characters on the adele ring

Definition. Let be a locally compact abelian group. The character group of is the set of all characters of and is denoted by Equivalently is the set of all continuous group homomorphisms from to We equip with the topology of uniform convergence on compact subsets of One can show that is also a locally compact abelian group.
Proof. By reduction to local coordinates it is sufficient to show each is self-dual. This can done by using a fixed character of We illustrate this idea by showing is self-dual. Define:
Then the following map is an isomorphism which respects topologies:

Tate's thesis

With the help of the characters of we can do Fourier analysis on the adele ring. John Tate in his thesis "Fourier analysis in number fields and Heckes Zeta functions" proved results about Dirichlet L-functions 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 zeta functions and the L-functions. We can define adelic forms of these functions and we can represent them as integrals over the adele ring or the idele group, with respect to corresponding Haar measures. We can show functional equations and meromorphic continuations of these functions. For example, for all with
where is the unique Haar measure on normalized such that has volume one and is extended by zero to the finite adele ring. As a result the Riemann zeta function can be written as an integral over the adele ring.

Automorphic forms

The theory of automorphic forms is a generalization of Tate's thesis by replacing the idele group with analogous higher dimensional groups. To see this note:
Based on these identification a natural generalization would be to replace the idele group and the 1-idele with:
And finally
where is the centre of Then we define an automorphic form as an element of In other words an automorphic form is a functions on satisfying certain algebraic and analytic conditions. For studying automorphic forms, it is important to know the representations of the group It is also possible to study automorphic L-functions, which can be described as integrals over
We could generalize even further by replacing with a number field and with an arbitrary reductive algebraic group.

Further applications

A generalisation of Artin reciprocity law leads to the connection of representations of and of Galois representations of .
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 global 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.