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Adele ring

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In mathematics, the adele ring of a global field (also adelic ring, ring of adeles or ring of adèles[1]) is a central object of class field theory, a branch of algebraic number theory. It is the restricted product of all the completions of the global field and is an example of a self-dual topological ring.

An adele derives from a particular kind of idele. "Idele" derives from the French "idèle" and was coined by the French mathematician Claude Chevalley. The word stands for 'ideal element' (abbreviated: id.el.). Adele (French: "adèle") stands for 'additive idele' (that is, additive ideal element).

The ring of adeles allows one to describe the Artin reciprocity law, which is a generalisation of quadratic reciprocity, and other reciprocity laws over finite fields. In addition, it is a classical theorem from Weil that -bundles on an algebraic curve over a finite field can be described in terms of adeles for a reductive group . Adeles are also connected with the adelic algebraic groups and adelic curves.

The study of geometry of numbers over the ring of adeles of a number field is called adelic geometry.

Definition

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Let be a global field (a finite extension of or the function field of a curve over a finite field). The adele ring of is the subring

consisting of the tuples where lies in the subring for all but finitely many places . Here the index ranges over all valuations of the global field , is the completion at that valuation and the corresponding valuation ring.[2]

Motivation

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The ring of adeles solves the technical problem of "doing analysis on the rational numbers ." The classical solution was to pass to the standard metric completion and use analytic techniques there.[clarification needed] But, as was learned later on, there are many more absolute values other than the Euclidean distance, one for each prime number , as was classified by Ostrowski. The Euclidean absolute value, denoted , is only one among many others, , but the ring of adeles makes it possible to comprehend and use all of the valuations at once. This has the advantage of enabling analytic techniques while also retaining information about the primes, since their structure is embedded by the restricted infinite product.

The purpose of the adele ring is to look at all completions of at once. The adele ring is defined with the restricted product, rather than the Cartesian product. There are two reasons for this:

  • For each element of 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, there cannot be any application of harmonic analysis to the Cartesian product. This is because local compactness ensures the existence (and uniqueness) of Haar measure, a crucial tool in analysis on groups in general.

Why the restricted product?

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The restricted infinite product is a required technical condition for giving the number field a lattice structure inside of , making it possible to build a theory of Fourier analysis (cf. Harmonic analysis) in the adelic setting. This is analogous to the situation in algebraic number theory where the ring of integers of an algebraic number field embeds

as a lattice. With the power of a new theory of Fourier analysis, Tate was able to prove a special class of L-functions and the Dedekind zeta functions were meromorphic on the complex plane. Another natural reason for why this technical condition holds can be seen by constructing the ring of adeles as a tensor product of rings. If defining the ring of integral adeles as the ring

then the ring of adeles can be equivalently defined as

The restricted product structure becomes transparent after looking at explicit elements in this ring. The image of an element inside of the unrestricted product is the element

The factor lies in whenever is not a prime factor of , which is the case for all but finitely many primes .[3]

Origin of the name

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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" (French: adèle) stands for additive idele. Thus, an adele is an additive ideal element.

Examples

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Ring of adeles for the rational numbers

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The rationals have a valuation for every prime number , with , and one infinite valuation with . Thus an element of

is a real number along with a p-adic rational for each of which all but finitely many are p-adic integers.

Ring of adeles for the function field of the projective line

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Secondly, take the function field of the projective line over a finite field. Its valuations correspond to points of , i.e. maps over

For instance, there are points of the form . In this case is the completed stalk of the structure sheaf at (i.e. functions on a formal neighbourhood of ) and is its fraction field. Thus

The same holds for any smooth proper curve over a finite field, the restricted product being over all points of .

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The group of units in the adele ring is called the idele group

.

The quotient of the ideles by the subgroup is called the idele class group

The integral adeles are the subring

Applications

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Stating Artin reciprocity

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The Artin reciprocity law says that for a global field ,

where is the maximal abelian algebraic extension of and means the profinite completion of the group.

Giving adelic formulation of Picard group of a curve

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If is a smooth proper curve then its Picard group is[4]

and its divisor group is . Similarly, if is a semisimple algebraic group (e.g. , it also holds for ) then Weil uniformisation says that[5]

Applying this to gives the result on the Picard group.

Tate's thesis

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There is a topology on for which the quotient is compact, allowing one to do harmonic analysis on it. John Tate in his thesis "Fourier analysis in number fields and Hecke Zeta functions"[6] 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.

Proving Serre duality on a smooth curve

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If is a smooth proper curve over the complex numbers, one can define the adeles of its function field exactly as the finite fields case. John Tate proved[7] that Serre duality on

can be deduced by working with this adele ring . Here L is a line bundle on .

Notation and basic definitions

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Global fields

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

Valuations

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For a valuation of it can be written for the completion of with respect to If is discrete it can be written for the valuation ring of and for the maximal ideal of If this is a principal ideal denoting the uniformising element by A non-Archimedean valuation is written as or and an Archimedean valuation as Then 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 (or absolute values) of Places corresponding to non-Archimedean valuations are called finite, whereas places corresponding to Archimedean valuations are called infinite. Infinite places of a global field form a finite set, which is denoted by

Define and let be its group of units. Then

Finite extensions

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Let be a finite extension of the global field Let be a place of and a place of If the absolute value restricted to is in the equivalence class of , then lies above which is denoted by and defined as:

(Note that both products are finite.)

If , can be embedded in Therefore, is embedded diagonally in With this embedding is a commutative algebra over with degree

The adele ring

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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 (finite) 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, it is written as

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.

Lemma. There is a natural embedding of into given by the diagonal map:

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 it is regarded 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

the result is:

The adele ring of rationals

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By Ostrowski's theorem the places of are it is possible to 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

the difference between restricted and unrestricted product topology can be illustrated using a sequence in :

Lemma. Consider the following sequence in :
In the product topology this converges to , but it does not converge at all in the restricted product topology.

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 it has: 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

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Definition (profinite integers). The profinite integers are defined as the profinite completion of the rings with the partial order i.e.,

Lemma.

Proof. This follows from the Chinese Remainder Theorem.

Lemma.

Proof. 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 It must be the case that factors through uniquely, i.e., there exists a unique -linear map such that can be defined 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.

Corollary. Define This results in an algebraic isomorphism

Proof.

Lemma. For a number field

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

The adele ring of a finite extension

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If be a finite extension, then is a global field. Thus is defined, and 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

Lemma. If is a finite extension, then both algebraically and topologically.

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

Furthermore, there are the following isomorphisms:

For the second use the map:

in which is the canonical embedding and The restricted product is taken on both sides with respect to

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

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

The adele ring of vector-spaces and algebras

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

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

Now, an alternative characterisation of the adele ring can be presented. 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

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Let be a finite dimensional vector-space over and a basis for over For each place of :

The adele ring of is defined as

This definition is based on the alternative description of the adele ring as a tensor product equipped with the same topology that was defined when giving an alternate definition of adele ring for number fields. Next, is equipped with the restricted product topology. Then and is embedded in naturally via the map

An alternative definition of the topology on can be provided. 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.

The topology can be defined in a different way. Fixing a basis for over results in an isomorphism Therefore fixing a basis induces an isomorphism The left-hand side is supplied 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, a linear homeomorphism which transfers the two topologies into each other is obtained. 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

[9]

The adele ring of an algebra

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Let be a finite-dimensional algebra over In particular, is a finite-dimensional vector-space over As a consequence, is defined and Since there is multiplication on and a multiplication on can be defined 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 , is defined as the -module generated by in For each finite set of places, 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

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Let be a finite extension. Since and from the Lemma above, can be interpreted as a closed subring of For this embedding, write . 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. Define depending on via the equations:

Now, 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 the result is:

Moreover, it can be proven that:[10]

Properties of the adele ring

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Theorem.[11] For every set of places is a locally compact topological ring.

Remark. The result above also holds for the adele ring of vector-spaces and algebras over

Theorem.[12] is discrete and cocompact in In particular, is closed in

Proof. 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 It is claimed that Let then and for all and therefore Additionally, and therefore Next, to show compactness, define:

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, it can be claimed that:

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, it can be deduced that there exists such that Now select such that Then The continuous projection is surjective, therefore as the continuous image of a compact set, is compact.

Corollary. Let be a finite-dimensional vector-space over Then is discrete and cocompact in
Theorem. The following are assumed:
  • is a divisible group.[13]
  • is dense.

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 because the finite number of denominators in the coordinates of the elements of can be reached 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, it can be assumed that

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 ( is well-defined, because only finitely many primes divide ). 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

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

Theorem.[14] Since is a locally compact group with addition, there is an additive Haar measure on This measure can be normalised such that every integrable simple function satisfies:
where for is the measure on such that has unit measure and is the Lebesgue measure. The product is finite, i.e., almost all factors are equal to one.

The idele group

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Definition. 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. is equipped 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 it can be assumed:

Since for all for large enough. However, as was seen 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. It is necessary to show is open or equivalently, that is open. But this is the condition above.

The idele group is equipped with the topology defined in the Lemma making it a topological group.

Definition. For a subset of places of set:

Lemma. The following identities of topological groups hold:
where 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. Prove the identity for ; the other two follow similarly. First 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, it is possible to 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.

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

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, 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 equation, is the finite idele group, meaning the group of units of the finite adele ring. For the infinite places, use the multiplicative lebesgue measure

The idele group of a finite extension

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Lemma. Let be a finite extension. Then:
where the restricted product is with respect to
Lemma. There is a canonical embedding of in

Proof. 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 algebras

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[15]

The idele group of an algebra

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Let be a finite-dimensional algebra over Since is not a topological group with the subset-topology in general, 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 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 [16]

Alternative characterisation of the idele group

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Proposition. Let be a finite set of places. Then
is an open subgroup of where is the union of all [17]
Corollary. In the special case of for each finite set of places
is an open subgroup of Furthermore, is the union of all

Norm on the idele group

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The trace and the norm should be transfer 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, it can be said that 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

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Lemma. There is natural embedding of into given by the diagonal map:

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

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.

Lemma.[18] Let be a finite extension. The embedding induces an injective map:

Properties of the idele group

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Absolute value on the idele group of K and 1-idele

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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 will be used to denote both the function on and

Theorem. is a continuous group homomorphism.

Proof. Let

where it is used 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. The set of -idele can be defined as:

is a subgroup of Since it is a closed subset of Finally the -topology on equals the subset-topology of on [19][20]

Artin's Product Formula. for all

Proof.[21] Proof of the formula for number fields, the case of global function fields can be proved similarly. Let be a number field and It has to be shown that:

For finite place for which the corresponding prime ideal does not divide , and therefore This is valid for almost all There is:

In going from line 1 to line 2, the identity was used where is a place of and is a place of lying above Going from line 2 to line 3, a property of the norm is used. The norm is in so without loss of generality it can be assumed that 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:

Lemma.[22] There exists a constant depending only on such that for every satisfying there exists such that for all
Corollary. Let be a place of and let be given for all with the property for almost all Then there exists so that for all

Proof. Let be the constant from the lemma. Let be a uniformising 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

Theorem. is discrete and cocompact in

Proof.[23] 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. It can be claimed that 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.

Theorem.[24] There is a canonical isomorphism Furthermore, is a set of representatives for and is a set of representatives for

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 it can be seen that proves injectivity. To show surjectivity let The absolute value of this element is and therefore

Hence and there is:

Since

It has been concluded that is surjective.

Theorem.[24] The absolute value function induces the following isomorphisms of topological groups:

Proof. The isomorphisms are given by:

Relation between ideal class group and idele class group

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Theorem. Let be a number field with ring of integers group of fractional ideals and ideal class group Here's the following isomorphisms
where has been defined.

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, both sides are divided 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, the embedding of into is used. In line 2, the definition of the map is used. Finally, 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, it has to be shown that 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 The second isomorphism of the theorem has been proven.

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 the idele group and idele class group of K

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Theorem.

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 It is defined that and furthermore, for and for Define It stands, that Therefore, is surjective.

The other equations follow similarly.

Characterisation of the idele group

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Theorem.[25] Let be a number field. There exists a finite set of places such that:

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 immediate, so the statement has been proved 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

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Finiteness of the class number of a number field

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In the previous section the fact that the class number of a number field is finite had been used. Here this statement can be proved:

Theorem (finiteness of the class number of a number field). Let be a number field. Then

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.[26]

Group of units and Dirichlet's unit theorem

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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

Lemma 1. Let The following set is finite:

Proof. Define

is compact and the set described above is the intersection of with the discrete subgroup in and therefore finite.

Lemma 2. Let be set of all such that for all Then the group of all roots of unity of In particular it is finite and cyclic.

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

Unit Theorem. is the direct product of and a group isomorphic to where if and if [27]
Dirichlet's Unit Theorem. Let be a number field. Then where is the finite cyclic group of all roots of unity of is the number of real embeddings of and is the number of conjugate pairs of complex embeddings of It stands, that

Remark. The Unit Theorem generalises Dirichlet's Unit Theorem. To see this, let be a number field. It is already known that set and note

Then there is:

Approximation theorems

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Weak Approximation Theorem.[28] Let be inequivalent valuations of Let be the completion of with respect to Embed diagonally in Then is everywhere dense in In other words, for each and for each there exists such that:
Strong Approximation Theorem.[29] Let be a place of Define
Then is dense in

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

Hasse principle

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Hasse-Minkowski Theorem. A quadratic form on 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. The idea of the Hasse principle (also known as local–global 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

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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 Equip with the topology of uniform convergence on compact subsets of One can show that is also a locally compact abelian group.

Theorem. The adele ring is self-dual:

Proof. By reduction to local coordinates, it is sufficient to show each is self-dual. This can be done by using a fixed character of The idea has been illustrated by showing is self-dual. Define:

Then the following map is an isomorphism which respects topologies:

Theorem (algebraic and continuous duals of the adele ring).[30] Let be a non-trivial character of which is trivial on Let be a finite-dimensional vector-space over Let and be the algebraic duals of and Denote the topological dual of by and use and to indicate the natural bilinear pairings on and Then the formula for all determines an isomorphism of onto where and Moreover, if fulfils for all then

Tate's thesis

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With the help of the characters of Fourier analysis can be done on the adele ring.[31] John Tate in his thesis "Fourier analysis in Number Fields and Hecke Zeta Functions"[6] 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. Adelic forms of these functions can be defined and represented as integrals over the adele ring or the idele group, with respect to corresponding Haar measures. Functional equations and meromorphic continuations of these functions can be shown. For example, for all with

where is the unique Haar measure on normalised 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 (a subset of) the adele ring.[32]

Automorphic forms

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The theory of automorphic forms is a generalisation of Tate's thesis by replacing the idele group with analogous higher dimensional groups. To see this note:

Based on these identification a natural generalisation would be to replace the idele group and the 1-idele with:

And finally

where is the centre of Then it define an automorphic form as an element of In other words an automorphic form is a function 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 [33]

Generalise even further is possible by replacing with a number field and with an arbitrary reductive algebraic group.

Further applications

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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 global field. The Artin reciprocity law, which is a sweeping generalisation of the Gauss quadratic reciprocity law, states that the product vanishes on the multiplicative group of the number field. Thus, the global reciprocity map of the idele class group to the abelian part of the absolute Galois group of the field will be obtained.

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.

References

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  1. ^ Groechenig, Michael (August 2017). "Adelic Descent Theory". Compositio Mathematica. 153 (8): 1706–1746. arXiv:1511.06271. doi:10.1112/S0010437X17007217. ISSN 0010-437X. S2CID 54016389.
  2. ^ Sutherland, Andrew (1 December 2015). 18.785 Number theory I Lecture #22 (PDF). MIT. p. 4.
  3. ^ "ring of adeles in nLab". ncatlab.org.
  4. ^ Geometric Class Field Theory, notes by Tony Feng of a lecture of Bhargav Bhatt (PDF).
  5. ^ Weil uniformization theorem, nlab article.
  6. ^ a b Cassels & Fröhlich 1967.
  7. ^ Tate, John (1968), "Residues of differentials on curves" (PDF), Annales Scientifiques de l'École Normale Supérieure, 1: 149–159, doi:10.24033/asens.1162.
  8. ^ This proof can be found in Cassels & Fröhlich 1967, p. 64.
  9. ^ The definitions are based on Weil 1967, p. 60.
  10. ^ See Weil 1967, p. 64 or Cassels & Fröhlich 1967, p. 74.
  11. ^ For proof see Deitmar 2010, p. 124, theorem 5.2.1.
  12. ^ See Cassels & Fröhlich 1967, p. 64, Theorem, or Weil 1967, p. 64, Theorem 2.
  13. ^ The next statement can be found in Neukirch 2007, p. 383.
  14. ^ See Deitmar 2010, p. 126, Theorem 5.2.2 for the rational case.
  15. ^ This section is based on Weil 1967, p. 71.
  16. ^ A proof of this statement can be found in Weil 1967, p. 71.
  17. ^ A proof of this statement can be found in Weil 1967, p. 72.
  18. ^ For a proof see Neukirch 2007, p. 388.
  19. ^ This statement can be found in Cassels & Fröhlich 1967, p. 69.
  20. ^ is also used for the set of the -idele but is used in this example.
  21. ^ There are many proofs for this result. The one shown below is based on Neukirch 2007, p. 195.
  22. ^ For a proof see Cassels & Fröhlich 1967, p. 66.
  23. ^ This proof can be found in Weil 1967, p. 76 or in Cassels & Fröhlich 1967, p. 70.
  24. ^ a b Part of Theorem 5.3.3 in Deitmar 2010.
  25. ^ The general proof of this theorem for any global field is given in Weil 1967, p. 77.
  26. ^ For more information, see Cassels & Fröhlich 1967, p. 71.
  27. ^ A proof can be found in Weil 1967, p. 78 or in Cassels & Fröhlich 1967, p. 72.
  28. ^ A proof can be found in Cassels & Fröhlich 1967, p. 48.
  29. ^ A proof can be found in Cassels & Fröhlich 1967, p. 67
  30. ^ A proof can be found in Weil 1967, p. 66.
  31. ^ For more see Deitmar 2010, p. 129.
  32. ^ A proof can be found Deitmar 2010, p. 128, Theorem 5.3.4. See also p. 139 for more information on Tate's thesis.
  33. ^ For further information see Chapters 7 and 8 in Deitmar 2010.

Sources

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