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. Formally, the idele group of a global field is the restricted direct product
of the multiplicative groups of the completions of , taken with respect to the unit groups at the non-archimedean places. Equivalently, it is the group of invertible elements of the adele ring , equipped with a topology finer than the subspace topology inherited from .
The quotient 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 -functions and Hecke characters.
Definition
Let be a global field, and let run over the places of . For each place , let denote the completion of at . If is non-archimedean, let be the corresponding valuation ring and let be its group of units.
The idele group of , usually denoted or , is the restricted product
of the groups , taken with respect to the subgroups at the non-archimedean places. Thus an idele is a family
such that
for all but finitely many non-archimedean places . Multiplication is defined componentwise.
Equivalently, the idele group is the group of invertible elements of the adele ring . However, its topology is not the subspace topology inherited from ; it is the restricted product topology, or equivalently the topology induced by the embedding
The multiplicative group embeds diagonally in . The quotient
is called the idele class group of .
Motivation
The idele group may be viewed as a topological refinement of the group of fractional ideals of a number field. If is a number field with ring of integers , every nonzero fractional ideal has a unique factorization
where runs over the nonzero prime ideals of and all but finitely many integers are zero. Thus the group of fractional ideals records, for each finite place of , an integral valuation.
An idele records similar local valuation data, but with additional local information. For an idele , the component at a finite place determines an integer . Since is a unit for all but finitely many , these integers define a fractional ideal
This gives a surjective homomorphism from the idele group to the group of fractional ideals. The diagonal embedding sends an element to the principal idele whose associated fractional ideal is the principal 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
Although the idele group is the group of invertible elements of the adele ring , it is not usually equipped with the subspace topology inherited from . With the subspace topology, inversion need not be continuous. Instead, is given the restricted product topology
where the restricted product is taken with respect to the compact open subgroups at the non-archimedean places. A basis of open neighbourhoods of the identity is given by products
where is an open neighbourhood of in and for all but finitely many non-archimedean places . Equivalently, this is the topology induced by the embedding
With this topology, is a locally compact topological group.
Since the idele group is locally compact, it has a Haar measure, usually denoted . This measure is obtained as a product of local multiplicative Haar measures on the groups . At a non-archimedean place , the local measure is commonly normalized so that
At the real place, a standard multiplicative Haar measure on is
up to multiplication by a positive constant; analogous normalizations are used at complex places. These local choices combine to give a multiplicative Haar measure on . Such measures are used in harmonic analysis on the ideles, especially in Tate's thesis and in the analytic theory of Hecke -functions.
Norm map and norm-one ideles
The idele group carries a homomorphism, usually called the idele norm or module into the positive reals. Choose the standard normalized absolute value on each completion : for a non-archimedean place , it is normalized so that
where is a uniformizer and is the size of the residue field. At the archimedean places one uses the usual normalized absolute values, with the complex absolute value taken squared. For an idele , define
This product is finite, since for all but finitely many non-archimedean places, and hence for all but finitely many . Thus
is a continuous group homomorphism.
The norm-one ideles are the elements in the kernel of this homomorphism:
By the product formula for global fields, every element of , embedded diagonally in , has idele norm one. Hence
The quotient
is called the group of norm-one idele classes. It is a compact group.
The idele norm descends to a homomorphism on the idele class group,
whose kernel is . For number fields this gives an exact sequence
Thus the idele class group is not compact in the number field case, but its norm-one subgroup modulo 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.
For number fields, the idele norm is surjective onto , and the above exact sequence splits after choosing a positive archimedean component. Thus is, non-canonically or after such a choice, a product of the compact group with . For global function fields, the image of the idele norm is instead a discrete subgroup of , so the corresponding quotient is discrete and isomorphic to an infinite cyclic group.
Norms for field extensions
Let be a finite extension of global fields. For each place of and each place of lying above , there is a local norm map
These local norm maps combine to give a continuous homomorphism on idele groups
If , then the -component of is
This product is finite for each fixed . Moreover, for all but finitely many non-archimedean places , the component lies in , and its local norm lies in . Hence is again an idele of . The continuity follows from the continuity of the local norm maps and from the restricted product topology.
The norm map is compatible with principal ideles. If is embedded diagonally in , then
is the principal idele of associated with the field norm . Consequently, the idele norm descends to a continuous homomorphism on idele class groups,
where and .
The embedding of into also gives a natural homomorphism
Explicitly, an idele of is sent to the idele whose component at is the image of in . Under this embedding,
where the power is taken componentwise. This follows from the identity
The field-extension norm should be distinguished from the idele norm or module . They are nevertheless compatible: with the standard normalized absolute values,
In particular, maps the norm-one idele group into and induces a homomorphism
In global class field theory, the image is called the norm subgroup of . For a finite abelian extension , the global Artin reciprocity map identifies the quotient
with the Galois group , up to the usual convention concerning arithmetic or geometric Frobenius.
Example: the rational numbers
For , the finite adele ring is
and the finite integral adeles are
The finite ideles are
where the restricted product is taken with respect to . The idele group of is
Every idele class has a representative of the form
Indeed, multiplying by a rational number changes the finite valuations and can be used to make all finite components -adic units; the remaining positive real factor records the idele norm. Thus
Similarly, the norm-one idele classes are
This reflects the fact that has trivial ideal class group: the remaining finite part of the idele class group comes from the local unit groups .
Class field theory
The idele class group yields a formulation of class field theory. Global class field theory describes the abelian extensions of a global field in terms of topological quotients of
The main result is the global Artin reciprocity law. In one formulation, for every finite abelian extension there is a canonical reciprocity homomorphism
whose kernel is the norm subgroup
The reciprocity homomorphism induces an isomorphism
up to a conventional choice of arithmetic or geometric Frobenius automorphism.
Thus, finite abelian extensions of correspond to open subgroups of finite index in the idele class group. Under this correspondence, an extension is associated with the subgroup . This replaces the older formulation of class field theory in terms of ideal class groups, ray class groups, and congruence conditions by a topological statement about quotients of .
The idelic formulation also incorporates the local reciprocity maps of local class field theory: for each place of , local class field theory relates to the abelianized Galois group of . 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 , the finite-level reciprocity maps are compatible as varies over finite abelian extensions of . They combine into a global reciprocity map from the idele class group to . Thus the abelianized absolute Galois group of is described by the system of finite quotients of the idele class group.
Hecke characters and L-functions
A Hecke character of a global field can be described as a continuous homomorphism
or equivalently as a continuous character of the idele group that is trivial on the diagonally embedded subgroup . Such characters are the automorphic characters of .
Writing an idele as , a Hecke character decomposes into local characters
with
For all but finitely many non-archimedean places , the local character is unramified, meaning that it is trivial on . At such a place its value is determined by , where is a uniformizer of .
Associated to a Hecke character is a global -function, defined for suitable by an Euler product
At an unramified non-archimedean place , the local factor has the form
where is the size of the residue field. The remaining finitely many finite places give ramified local factors, and the archimedean places contribute gamma factors. These local factors combine to form the completed Hecke -function.
Classical Dirichlet characters and ideal class characters occur as special cases. For example, over , 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.