Adele ring

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

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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: