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 -adic numbers for all prime numbers . More generally, if is a global field, its adele ring, often denoted , is a topological ring built from the completions of at all its places . Formally, it is a restricted product of the local fields , with respect to the valuation rings at the non-archimedean places. Its elements are called adeles.
The restricted product topology makes a locally compact topological ring. The field embeds diagonally in as a discrete subring, and the quotient 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 . The quotient , 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
Let be a global field, meaning either a number field or a global function field. Let run over the places of . For each place , let be the completion of at . If is non-archimedean, let be the corresponding valuation ring.
The set of finite adeles of , denoted , is the restricted product of the non-archimedean completions with respect to the subrings :
It is equipped with the restricted product topology. A basis of open sets is given by products
where is a finite set of non-archimedean places and each is open in . With componentwise addition and multiplication, is a topological ring.
The adele ring of , denoted , is obtained by adjoining the completions at the archimedean places:
The number of archimedean places is finite, and each archimedean completion is isomorphic to or . The elements of are called adeles of . 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.
If is a global function field, then there are no archimedean places, so .
There is a natural diagonal embedding
This map is well-defined because an element lies in for all but finitely many non-archimedean places . After this embedding, is regarded as a subring of , and its elements are sometimes called the principal adeles of .
More generally, if is a set of places of , one may define the ring of -adeles by
again using the valuation rings at the non-archimedean places in . If
then there is a natural product decomposition
Motivation
The purpose of the adele ring is to look at all completions of a global field at once. For the rational numbers, the usual absolute value gives the completion , but Ostrowski's theorem shows that there are also the -adic absolute values, one for each prime number . More generally, a global field has a family of completions , one for each place . 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. If is a number field with ring of integers , the classical Minkowski embedding places as a lattice in the finite-dimensional real vector space
This makes it possible to study arithmetic questions using volume and compactness arguments. The adele ring may be viewed as a local-global enlargement of this construction: instead of using only the archimedean completions, it includes all completions of . In the adelic setting, the global field itself embeds diagonally as a discrete subgroup of , and the quotient is compact.
The adele ring is defined as a restricted product rather than as the full Cartesian product of all completions. The restricted product condition says that an adele is integral at almost all non-archimedean places. This condition is natural from the point of view of the global field itself: if , then belongs to for all but finitely many finite places . Thus the diagonal embedding
lands in the restricted product.
The restricted product is also the topological condition that makes the adele ring useful for analysis. With its restricted product topology, is a locally compact topological ring. Local compactness gives the additive group of 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, for example, constructs Fourier analysis on the adele ring and integration over the idele group to give a uniform treatment of Hecke -functions. In this approach, global zeta integrals factor into local integrals over the completions , and the local-global structure of the adele ring explains the Euler product, analytic continuation, and functional equation of these -functions.
Examples
Rational adeles
For , Ostrowski's theorem says that the places of are given by the usual absolute value and the -adic absolute values, one for each prime number . The completion at the infinite place is
and the completion at the place corresponding to is the field of -adic numbers , with valuation ring . Thus the adele ring of is
where the restricted product is taken with respect to the subrings . Equivalently,
Thus an adele of is a real number together with a -adic rational number for each prime , such that all but finitely many of the -adic components are -adic integers.
The finite adeles of are
The integral finite adeles are
the ring of profinite integers. With this notation,
The diagonal embedding of sends a rational number to the adele
This is well-defined because a rational number has only finitely many prime factors in its denominator, so for all but finitely many primes .
Number fields
Let be a number field with ring of integers . At each finite place , the completion is a finite extension of some , and its valuation ring is denoted . At each infinite place, the completion is isomorphic to either or . The adele ring is
where the restricted product over finite places is taken with respect to the rings . Thus an adele of is a family with for every place , such that for all but finitely many finite places.
For example, if is a quadratic number field, then its archimedean factor is either , when has two real embeddings, or , when has one pair of complex embeddings. The finite part is a restricted product over the nonzero prime ideals of .
If is a finite extension of number fields, then the adelic construction is compatible with extension of scalars. In particular, one has a natural isomorphism
and, in the special case ,
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 over a finite field. Its places correspond to the closed points of . Such points may be described as maps
over . For instance, there are points of the form
For a point , the local ring used in the restricted product is the completed local ring
and the corresponding local field is its fraction field, often denoted . Thus the adele ring of may be written
where the restricted product is taken with respect to the completed local rings . Equivalently, its elements are families , with , such that for all but finitely many points .
The same description holds for any smooth proper curve over a finite field. If is its function field, then
where runs over the closed points of . 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
The topology on the adele ring is the restricted product topology. For a finite set of places containing the archimedean places, define
Equipped with the product topology and componentwise addition and multiplication, is a locally compact topological ring. If is another finite set of places of containing , then is an open subring of . The adele ring is the union of all these open subrings:
Equivalently, is the set of all such that for almost all non-archimedean places . The topology of is induced by the requirement that all be open subrings. Thus is a locally compact topological ring.
The same construction applies to sets of places. For every set of places , the ring of -adeles
is a locally compact topological ring, with the restricted product topology. If
then there is a natural product decomposition
The diagonal embedding
identifies with a subring of . With this embedding, the elements of are called principal adeles. The image of is discrete in , and the quotient
is compact. In particular, is closed in . This compactness property is one of the main reasons the adele ring is useful in harmonic analysis and in arithmetic applications.
The adele ring also separates naturally into any chosen local factor and the remaining factors. Fix a place of . Let be a finite set of places containing and , and define
Then
Furthermore, define
where runs through all finite sets containing . Then
via the map
Thus there is a natural embedding and a natural projection . The same construction works with any finite set of places in place of the single place .
Since is locally compact as an additive group, it has an additive Haar measure. This measure is used in harmonic analysis on global fields and is usually normalized as a product of local Haar measures. With the standard normalization at the non-archimedean places, the valuation ring has measure for almost all finite places.
Haar measure and Fourier analysis
Since is locally compact as an additive group, it has an additive Haar measure, usually denoted . This measure may be normalized as a product of local Haar measures on the completions . At a non-archimedean place , the local measure is commonly normalized so that the valuation ring has measure ; at the archimedean places one uses the usual Lebesgue measure on or .
A function is called simple if
where each is measurable and for almost all non-archimedean places . With the standard normalization, every integrable simple function satisfies
The product is finite in the sense that almost all factors are equal to .
Fourier analysis on the adele ring is based on the characters of its additive group. If is a locally compact abelian group, its character group is the group of all continuous homomorphisms from to
with the topology of uniform convergence on compact subsets. The adele ring is self-dual as a locally compact abelian group:
This is proved by reducing to the corresponding local statement for each completion . For example, the usual character
gives an isomorphism
Analogous local characters are used at the non-archimedean places, and their restricted product gives the global self-duality of .
After choosing a nontrivial additive character , the Fourier transform of a suitable function on is defined by
With a compatible choice of Haar measure, this Fourier transform satisfies the usual inversion and Plancherel formulas. One of the important features of the adelic setting is that global Fourier analysis factors into local Fourier analysis over the completions .