In number theory, the idele group is a way of packaging the multiplicative arithmetic of a global field at all of its completions at once, so that it contains the information of unique factorization as well as the data relating to units is the restricted direct product at the non-archimedean placess. Equivalently, it is the group of invertible elements of the adele ring.

is the idele class group. Ideles and idele class groups are used in class field theory. They were exploited by John Tate in his thesis to formulate global zeta and L-functions and Hecke characters.

Definition

be a global field denote the completion be its group of units.

The idele group, is the restricted product

at the non-archimedean places. Thus an idele is a family

such that

. Multiplication is defined componentwise.

Equivalently, the idele group is the group of invertible elements of the adele ring. However, its topology is not; it is the restricted product topology, or equivalently the topology induced by the embedding

. The quotient

is called the idele class group.

Motivation

The idele group may be viewed as a topological refinement of the group of fractional ideals is a number field with ring of integers, every nonzero fractional ideal has a unique factorization

, an integral valuation.

, these integers define a fractional ideal

. Consequently, passing to quotients gives a natural surjection

from the idele class group to the ordinary ideal class group.

Thus the idele class group enlarges the ideal class group. It extends the finite-prime data measured by fractional ideals with the unit groups at finite places and the multiplicative groups at the archimedean places. This additional topological information is important in class field theory and in the theory of Hecke characters, where characters of the idele class group replace characters defined only on ideal class groups or ray class groups.

Topology and Haar measure

is given the restricted product topology

at the non-archimedean places. A basis of open neighbourhoods of the identity is given by products

. Equivalently, this is the topology induced by the embedding

is a locally compact topological group.

Since the idele group is locally compact, it has a Haar measure, the local measure is commonly normalized so that

is

-functions.

Norm map and norm-one ideles

The idele group carries a homomorphism, usually called the idele norm or module

is a continuous group homomorphism.

By the product formula, has idele norm one. Hence

is called the group of norm-one idele classes. It is a compact group.

is compact. This compactness is one of the idelic forms of the finiteness of the ideal class group together with the structure theorem for units.

, so the corresponding quotient is discrete and isomorphic to an infinite cyclic group.

Norms for field extensions

, there is a local norm map

These local norm maps combine to give a continuous homomorphism on idele groups

is

. The continuity follows from the continuity of the local norm maps and from the restricted product topology.

, then

. Consequently, the idele norm descends to a continuous homomorphism on idele class groups,

.

also gives a natural homomorphism

. Under this embedding,

where the power is taken componentwise. This follows from the identity

. They are nevertheless compatible: with the standard normalized absolute values,

and induces a homomorphism

, the global Artin reciprocity map identifies the quotient

, up to the usual convention concerning arithmetic or geometric Frobenius.

Example: the rational numbers

, the finite adele ring is

and the finite integral adeles are

The finite ideles are

is

Every idele class has a representative of the form

-adic units; the remaining positive real factor records the idele norm. Thus

Similarly, the norm-one idele classes are

.

Class field theory

The idele class group yields a formulation of class field theory

The main result is the global Artin reciprocity law

up to a conventional choice of arithmetic or geometric Frobenius automorphism.

.

The idelic formulation also incorporates the local reciprocity maps of local class field theory to the abelianized Galois group. The global reciprocity map is compatible with these local maps through the embedding of each local multiplicative group into the idele group. At an unramified finite place, a uniformizer maps to a Frobenius element, with the precise inverse depending on the convention used for the Artin map.

Classical ideal-theoretic class field theory is then a special case. Quotients of the idele class group by certain subgroups recover ray class groups, and the corresponding abelian extensions are the ray class fields. In particular, the Hilbert class field is obtained from the quotient associated with the ordinary ideal class group, packaging the relation between ideles, fractional ideals, and ideal classes.

For the maximal abelian extension is described by the system of finite quotients of the idele class group.

Hecke characters and L-functions

A Hecke character can be described as a continuous homomorphism

or equivalently as a continuous character. Such characters are the automorphic characters.

, a Hecke character decomposes into local characters

with

.

by an Euler product

, the local factor has the form

is the size of the residue field. The remaining finitely many finite places give ramified local factors, and the archimedean places contribute gamma-function.

Classical Dirichlet characters, Dirichlet characters can be interpreted as finite-order Hecke characters with prescribed finite conductors. More generally, ray class characters of a number field can be realized as finite-order characters of quotients of the idele class group.

-functions. In Tate's thesis-functions are obtained by harmonic analysis-series, while also explaining their local-global factorization in terms of the product structure of the ideles.

Relation with the ideal class group

, and let

. Its group of units is

denote the finite idele group,

There is a natural surjective homomorphism

defined by

. Hence

This identifies the group of fractional ideals with the quotient of the finite idele group obtained by forgetting the local unit components.

. Therefore the preceding homomorphism descends to a quotient map from finite idele classes to ideal classes. In particular,

Equivalently, using the full idele group,

Thus the ordinary ideal class group is obtained from the idele class group by quotienting out the finite local unit groups and the archimedean multiplicative factors.

and also discards the archimedean components. These extra local and topological data are precisely what make the idele class group suitable for class field theory and for the theory of Hecke characters.

identifies principal fractional ideals with principal ideles, giving the ideal class group.

Further structure and proof sketches

The following standard structural facts give equivalent descriptions of the idele topology, related subgroups, and some compactness and decomposition results used in the arithmetic theory of ideles.

Topology induced from the adele ring

be a topological ring. Define

a topological group.

1. Proof.

is open, or equivalently that

. The idele group is equipped with this topology.

is not continuous. The sequence

; without loss of generality it can be assumed that

.

Subgroups attached to sets of places

, set

The following identities of topological groups hold:

Here the restricted product has the restricted product topology, generated by restricted open rectangles of the form

are open sets.

1. Proof.

; the other two follow similarly. First show the two sets are equal:

.

is the union of all these restricted open rectangles and is open in the restricted product topology.

as a restricted product, and the topological group property follows from the preceding discussion on the group of units of a topological ring.

is given by all sets of the form

.

Finite extensions

be a finite extension. Then

.

with the property

.

induces an injective map

Principal ideles and discreteness

given by the diagonal map

, the embedding is well-defined and injective. In analogy to the ideal class group are called principal ideles.

. Therefore

is a locally compact topological group and a Hausdorff space.

.

Product formula and compactness of norm-one idele classes

, define

is an idele, this product is finite and therefore well-defined. The set of norm-one ideles is

.

The product formula states that

.

1. Proof.

. It has to be shown that

. There is

In going from the first line to the second, the identity

. Write

. Then

The following approximation lemma is used in the proof of compactness.

1. Lemma. satisfying

such that

.

1. Corollary. such that

.

1. Proof.

minimal so that

, so that

such that

.

1. Theorem..

2. Proof.

satisfies

Define

is compact. It can be claimed that the natural projection

be arbitrary. Then

and therefore

It follows that

such that

. This proves the surjectivity of the natural projection. Since it is also continuous, compactness follows.

The rational numbers

There is a canonical isomorphism

.

1. Proof.

Consider the map

and therefore

is a continuous group homomorphism. Suppose

such that

, which proves injectivity. To show surjectivity, let

, and therefore

, and there is

Since

is surjective.

The absolute value function induces the following isomorphisms of topological groups:

The isomorphisms are given by

and

Decomposition of the idele group and idele class group

The idele norm gives the following decompositions:

1. Proof.

.

, define

. The isomorphism is given by multiplication:

. Since

is injective.

. Let

Then

follow similarly.

Characterisation by a finite set of places

such that

1. Proof.

. Consider the isomorphism

induced by

. The corresponding ideal

, meaning

. That means

.

Ideles of finite-dimensional algebras

.

.

be a finite set of places. Then

,

.

References

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