Wikipedia / Adele ring
Wikipedia / Adele ring
In number theory, the adele ring is a construction that combines all local versions of a global field into one object. For the rational numbers, these local versions include the real numbers and the fields of p-adic numbers for all prime numbers, is a topological ring at all its places. Formally, it is a restricted product, with respect to the valuation rings at the non-archimedean places. Its elements are called adeles.
The restricted product a locally compact is compact. As an additive locally compact abelian group, the adele ring is self-dual, making it a natural setting for Fourier analysis on global fields.
The group of units of the adele ring, with its natural topology, is the idele group, called the idele class group, is a central object in class field theory. Adeles and ideles are also used in Tate's thesis, the theory of automorphic forms, local-global principles, and adelic descriptions of divisors, line bundles, and principal bundles on algebraic curves.
Definition
be a global field, meaning either a number field be the completion be the corresponding valuation ring.
The set of finite adeles, is the restricted product:
It is equipped with the restricted product topology. A basis of open sets is given by products
is a topological ring.
The adele ring, is obtained by adjoining the completions at the archimedean places:
are called adeles. Addition and multiplication are defined componentwise. For brevity, one often writes
with the understanding that the restricted product condition applies only at the non-archimedean places.
.
There is a natural diagonal embedding
, and its elements are sometimes called the principal adeles.
, one may define the ring of -adeles by
. If
then there is a natural product decomposition
Motivation
, but Ostrowski's theorem. The adele ring packages these completions into a single object, so that analytic methods can be applied while still retaining arithmetic information from all finite primes.
A precursor to this point of view is Minkowski's geometry of numbers as a lattice in the finite-dimensional real vector space
is compact.
. Thus the diagonal embedding
lands in the restricted product.
a Haar measure, making it possible to do harmonic analysis on the adele ring. This is one of the main reasons adeles are useful in modern number theory.
Tate's thesis-functions.
Examples
Rational adeles
, Ostrowski's theorem. The completion at the infinite place is
is
. Equivalently,
-adic integers.
are
The integral finite adeles are
the ring of profinite integers. With this notation,
to the adele
.
Number fields
. The adele ring is
for all but finitely many finite places.
.
is a finite extension of number fields, then the adelic construction is compatible with extension of scalars. In particular, one has a natural isomorphism
,
This gives another way to view the adele ring of a number field as the adelic extension of the rational adele ring.
Function fields and curves
Now take the function field
of the projective line. Such points may be described as maps
points of the form
, the local ring used in the restricted product is the completed local ring
may be written
.
is its function field, then
. Unlike number fields, global function fields have no archimedean places, so the finite adele ring and the full adele ring are the same.
Topology and main properties
containing the archimedean places, define
. The adele ring is the union of all these open subrings:
is a locally compact topological ring.
-adeles
is a locally compact topological ring, with the restricted product topology. If
then there is a natural product decomposition
The diagonal embedding
, and the quotient
. This compactness property is one of the main reasons the adele ring is useful in harmonic analysis and in arithmetic applications.
, and define
Then
Furthermore, define
. Then
via the map
.
for almost all finite places.
Haar measure and Fourier analysis
is locally compact as an additive group, it has an additive Haar measure.
is called simple if
. With the standard normalization, every integrable simple function satisfies
.
to
with the topology of uniform convergence on compact subsets. The adele ring is self-dual as a locally compact abelian group:
. For example, the usual character
gives an isomorphism
.
is defined by
.
, Fourier analysis can be done on the adele ring. In Tate's thesis, John Tate used Fourier analysis on the adele ring and integration over the idele group to study the Riemann zeta function-functions. Adelic forms of these functions can be represented as integrals over the adele ring or the idele group, with respect to corresponding Haar measures. Their functional equations and meromorphic continuations can then be proved by applying Fourier analysis and Poisson summation in the adelic setting.
,
, and extended by zero to the finite adele ring. Thus the Riemann zeta function can be written as an integral over a subset of the adele ring.
Applications
Class field theory
The adele ring enters class field theory through its group of units, the idele group. The quotient
. In one formulation, the global Artin reciprocity law.
This adelic formulation packages the local reciprocity maps of local class field theory into a global statement. It replaces the older ideal-theoretic formulation, involving ideal class groups and ray class groups, by a statement about the topology and quotients of the idele class group.
Ideal classes and units
, the finite part of the idele group maps onto the group of fractional ideals by
The kernel is the product of the local unit groups. Consequently, the ordinary ideal class group can be recovered as a quotient of the idele class group. This viewpoint gives an adelic interpretation of the finiteness of the class number: the compactness of the norm-one idele classes implies that the ideal class group is compact, and since it is discrete, it is finite.
, Dirichlet's unit theorem states that
is the number of conjugate pairs of complex embeddings.
Tate's thesis and L-functions
, Fourier analysis can be done on the adele ring; integration over the idele group then gives zeta integrals.
In Tate's thesis, John Tate used Fourier analysis on the adele ring and the idele group to study the Riemann zeta function-functions. Adelic forms of these functions can be represented as integrals over the adele ring or the idele group, with respect to corresponding Haar measures. Their functional equations and meromorphic continuations can then be proved using Fourier analysis and Poisson summation in the adelic setting.
, one has an adelic integral representation of the Riemann zeta function,
, and extended by zero to the finite adele ring. Thus the Riemann zeta function can be written as an integral over a subset of the adele ring.
Automorphic forms
Adeles also provide the natural language for automorphic forms may be viewed as functions on
-functions can often be described by integrals over adelic groups.
is central in the modern theory of automorphic representations. This viewpoint is also one of the starting points of the Langlands program, which relates automorphic representations of adelic groups to Galois representations.
Approximation and local-global principles
The adele ring provides a unified interpretation of approximation theorems and local-global questions. The weak approximation theorem is dense in the product of the corresponding completions. The strong approximation theorem is dense in the restricted product over all other places. Thus the global field is discrete in its full adele ring, but becomes dense when one place is omitted.
Adelic language is also used to formulate local-global principles, such as the Hasse principle. The adele ring provides a single space in which these local conditions can be collected and studied together.
Curves, divisors, and bundles
For a smooth proper curve. In this setting, ideles recover the divisor and Picard groups of the curve. One has
and
Thus the divisor-class description of line bundles on a curve can be expressed adelically.
More generally, for an algebraic group, adelic double quotients describe moduli of bundles on curves. In Weil uniformization, for suitable groups such as semisimple groups, one has an adelic description of the form
, this recovers the adelic description of the Picard group.
Serre duality on curves
in a way analogous to the function-field case over finite fields. Tate proved that Serre duality,
.
Idele group
. It is usually denoted
. Equivalently, it is the restricted direct product
.
. Instead it is given the restricted product topology, equivalently the topology induced by the embedding
is an abelian locally compact topological group.
gives the subgroup of principal ideles. The quotient
is the idele class group. This group is a central object in class field theory.
The idele group also carries a natural absolute value, or module,
where the local absolute values are normalized in the standard way. The product is finite for ideles, since almost all finite components are units. The subgroup
is compact.
For number fields, the finite part of the idele group maps naturally onto the group of fractional ideals by
, so the ordinary ideal class group is recovered as a quotient of the idele class group. In this way, the idele class group refines the ideal class group by retaining local unit data and archimedean information.
Ideles are also used in harmonic analysis on global fields. In Tate's thesis-functions, including their Euler products, analytic continuation, and functional equations.
Further properties and proof sketches
The preceding sections give the basic definition and main uses of the adele ring. This section records some standard structural facts and proof sketches.
Restricted product topology
.
Lemma.:
, but it does not converge at all in the restricted product topology.
Proof. are finite subsets of the set of all places.
Alternative descriptions for number fields
The profinite integers are defined as the profinite completion i.e.,
Lemma.
Proof. This follows from the Chinese Remainder Theorem.
Lemma.
Proof. Use the universal property-bilinear function
-bilinear map and is unique with these properties.
Corollary.
Proof.
Lemma.
Remark.
Finite extensions
Lemma. both algebraically and topologically.
is given by
Proof.
Furthermore, there are the following isomorphisms:
For the second use the map
Corollary. summands.
Adeles of vector spaces and algebras
:
is defined as
is the coarsest topology for which all these extensions are continuous.
The left-hand side is supplied with the product topology and this topology is transported with the isomorphism onto the right-hand side. The topology does not 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,
can be defined via
define
, the definition above is consistent with the definition of the adele ring.
Trace and norm
,
be a tower of global fields. Then
be a basis of the field extension, via the equations
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:
, the result is
Moreover, it can be proven that:
Discreteness and compactness of the diagonal image
Theorem.
Proof. which contains no other rational number. The general case follows via translation. Define
Next, to show compactness, define
be another prime. Then
Next, it can be claimed that
, as the continuous image of a compact set, is compact.
Corollary.
Approximation theorems
Weak Approximation Theorem. is everywhere dense such that
Strong Approximation Theorem. Define
Remark.
Arithmetic consequences
Theorem (finiteness of the class number of a number field).
Proof. The map
is compact. In addition, it is discrete and so finite.
Remark. by the set of the principal divisors is a finite group.
be a finite set of places. Define
defined by
Lemma. The following set is finite:
Proof. Define
and therefore finite.
Lemma. In particular it is finite and cyclic.
Proof.
Unit Theorem.
Dirichlet's Unit Theorem. be a number field. Then
Remark. Then there is
Duality for adelic vector spaces
The self-duality of the adele ring extends to adelic vector spaces.
Theorem (algebraic and continuous duals of the adele ring). Then the formula
Notes
References
External links