Adele ring

In mathematics, the adele ring of a global field 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'. Adele stands for 'additive idele'.
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 Torsor |. 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

Let be a global 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.

Motivation

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. But, as was learned later on, there are many more absolute values other than the Euclidean distance, one for each prime number, as classified by Ostrowski's theorem. The Euclidean absolute value, denoted, is only one among many others,, but the ring of adeles makes it possible to comprehend and. 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 of Haar measure, a crucial tool in analysis on groups in general.
Why the restricted product?

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 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.

Origin of the name

The term "idele" is an invention of the French mathematician Claude Chevalley and stands for "ideal element". The term "adele" stands for additive idele. Thus, an adele is an additive ideal element.

Examples

Ring of adeles for the rational numbers

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

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 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.

Related notions

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

Stating Artin reciprocity

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

If is a smooth proper curve then its Picard group is
and its divisor group is. Similarly, if is a semisimple algebraic group then Weil uniformisation says that
Applying this to gives the result on the Picard group.

Tate's thesis

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" 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

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

Notation and basic definitions

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 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 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

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:
If, can be embedded in Therefore, is embedded 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, 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.
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

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 :
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.