Adele ring

In mathematics, the adele ring of a global field (also adelic ring, ring of adeles or ring of adèles) 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.
The ring of adeles allows one to elegantly describe the Artin reciprocity law, which is a vast generalization of quadratic reciprocity, and other reciprocity laws over finite fields. In addition, it is a classical theorem from Weil that formula_1-bundles on an algebraic curve over a finite field can be described in terms of adeles for a reductive group formula_1.
Definition.
Let formula_3 be a global field (a finite extension of formula_4 or the function field of a curve "X/Fq" over a finite field). The adele ring of formula_3 is the subring
consisting of the tuples formula_7 where formula_8 lies in the subring formula_9 for all but finitely many places formula_10. Here the index formula_10 ranges over all valuations of the global field formula_3, formula_13 is the completion at that valuation and formula_14 the valuation ring.
Motivation.
The ring of adeles solves the technical problem of "doing" analysis on the rational numbers formula_4. The "classical" solution used by people before was to pass to the standard metric completion formula_16 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 formula_17, as was classified by Ostrowski. Since the Euclidean absolute value, denoted formula_18, is only one among many others, formula_19, the ring of adeles makes it possible to have a compromise and use all of the valuations at once. This has the advantage of getting access to analytic techniques, while also retaining information about the primes since their structure is embedded by the restricted infinite product.
Why the restricted product?
The restricted infinite product is a required technical condition for giving the number field formula_4 a lattice structure inside of formula_21, making it possible to build a theory of Fourier analysis in the adelic setting. This is entirely analogous to the situation in algebraic number theory where the ring of integers of an algebraic number field embedsformula_22as 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 directly by constructing the ring of adeles as a tensor product of rings. If we define the ring of integral adeles formula_23 as the ringformula_24then the ring of adeles can be equivalently defined asformula_25The restricted product structure becomes transparent after looking at explicit elements in this ring. If we take a rational number formula_26 we find formula_27. For any tuple formula_28 we have the following series of equalitiesformula_29Then, for any formula_30 we still have formula_31 for formula_32, but for formula_33 formula_34 since there is an inverse power of formula_35. This shows any element in this new ring of adeles can have an element in formula_36 at only finitely many places formula_35.
Origin of the name.
In local class field theory, the group of units of the local field plays a central role. In global class field theory, the idele class group takes this role. The term "idele" () is an invention of the French mathematician Claude Chevalley (1909–1984) and stands for "ideal element" (abbreviated: id.el.). The term "adele" () stands for additive idele.
The idea of the adele ring is to look at all completions of formula_3 at once. At first glance, the Cartesian product could be a good candidate. However, the adele ring is defined with the restricted product. There are two reasons for this: 
Examples.
Ring of adeles for the rational numbers.
The rationals "K=Q " have a valuation for every prime number "p", with (Kν,Oν)=(Qp,Zp), and one infinite valuation "∞" with Q∞=R. Thus an element of
is a real number along with a "p"-adic rational for each "p" 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 "K=Fq(P1)=Fq(t)" of the projective line over a finite field. Its valuations correspond to points "x" of "X"=P1, i.e. maps over Spec Fq
For instance, there are "q+1" points of the form SpecFq → P1. In this case "Oν=ÔX,x" is the completed stalk of the structure sheaf at "x" (i.e. functions on a formal neighbourhood of "x") and "Kν=KX,x" is its fraction field. Thus
The same holds for any smooth proper curve "X/Fq" over a finite field, the restricted product being over all points of "x∈X".
Related notions.
The group of units in the adele ring is called the idele group
The quotient of the ideles by the subgroup "K×⊆IK" 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 "K",
where "Kab" is the maximal abelian algebraic extension of "K" and formula_47 means the profinite completion of the group.
Giving adelic formulation of picard group of a curve.
If "X/Fq" is a smooth proper curve then its Picard group is
and its divisor group is "Div(X)"=AK×/OK×. Similarly, if "G" is a semisimple algebraic group (e.g. "SLn", it also holds for "GL"n) then the Weil uniformisation says that
Applying this to "G=Gm" gives the result on the Picard group.
Tate's thesis.
There is a topology on AK for which the quotient AK/"K" is compact, allowing one to do harmonic analysis on it. John Tate in his thesis "Fourier analysis in number fields and Heckes 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 "X" is a smooth proper curve "over the complex numbers", one can define the adeles of its function field C("X") exactly as the finite fields case. John Tate proved that Serre duality on "X"
can be deduced by working with this adele ring AC("X"). Here "L" is a line bundle on "X".
Notation and basic definitions.
Global fields.
Throughout this article, formula_3 is a global field, meaning it is either a number field (a finite extension of formula_52) or a global function field (a finite extension of formula_53 for formula_35 prime and formula_55). By definition a finite extension of a global field is itself a global field. 
Valuations.
For a valuation formula_56 of formula_3 we write formula_58 for the completion of formula_3 with respect to formula_60 If formula_56 is discrete we write formula_62 for the valuation ring of formula_58 and formula_64 for the maximal ideal of formula_65 If this is a principal ideal we denote the uniformizing element by formula_66 A non-Archimedean valuation is written as formula_67 or formula_68 and an Archimedean valuation as formula_69 We assume all valuations to be non-trivial.
There is a one-to-one identification of valuations and absolute values. Fix a constant formula_70 the valuation formula_56 is assigned the absolute value formula_72 defined as:
Conversely, the absolute value formula_74 is assigned the valuation formula_75 defined as: 
A place of formula_3 is a representative of an equivalence class of valuations (or absolute values) of formula_78 Places corresponding to non-Archimedean valuations are called finite, whereas places corresponding to Archimedean valuations are called infinite. The set of infinite places of a global field is finite, we denote this set by formula_79
Define formula_80 and let formula_81 be its group of units. Then formula_82
Finite extensions.
Let formula_83 be a finite extension of the global field formula_78 Let formula_85 be a place of formula_86 and formula_56 a place of formula_78 We say formula_85 lies above formula_90 denoted by formula_91 if the absolute value formula_92 restricted to formula_3 is in the equivalence class of formula_60 Define 
Note that both products are finite.
If formula_96 we can embed formula_58 in formula_98 Therefore, we can embed formula_58 diagonally in formula_100 With this embedding formula_101 is a commutative algebra over formula_58 with degree 
The adele ring.
The set of finite adeles of a global field formula_104 denoted formula_105 is defined as the restricted product of formula_58 with respect to the formula_107
It is equipped with the restricted product topology, the topology generated by restricted open rectangles, which have the following form:
where formula_110 is a finite set of (finite) places and formula_111 are open. With component-wise addition and multiplication formula_112 is also a ring. 
The adele ring of a global field formula_3 is defined as the product of formula_112 with the product of the completions of formula_3 at its infinite places. The number of infinite places is finite and the completions are either formula_116 or formula_117 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 formula_78 In the following, we write
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 formula_124 then formula_125 for almost all formula_60 This shows the map is well-defined. It is also injective because the embedding of formula_3 in formula_58 is injective for all formula_60 
Remark. By identifying formula_3 with its image under the diagonal map we regard it as a subring of formula_131 The elements of formula_3 are called the principal adeles of formula_131
Definition. Let formula_134 be a set of places of formula_78 Define the set of the formula_134-adeles of formula_3 as
Furthermore if we define 
we have: formula_140
The adele ring of rationals.
By Ostrowski's theorem the places of formula_52 are formula_142 where we identify a prime formula_35 with the equivalence class of the formula_35-adic absolute value and formula_145 with the equivalence class of the absolute value formula_18 defined as:
The completion of formula_52 with respect to the place formula_35 is formula_150 with valuation ring formula_151 For the place formula_145 the completion is formula_153 Thus: 
Or for short
We will illustrate the difference between restricted and unrestricted product topology using a sequence in formula_156: 
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 formula_160 and for each restricted open rectangle formula_161 we have: formula_162 for formula_163 and therefore formula_162 for all formula_165 As a result formula_166 for almost all formula_167 In this consideration, formula_110 and formula_169 are finite subsets of the set of all places.
Alternative definition for number fields.
Definition (profinite integers). We define profinite integers as the profinite completion of the rings formula_170 with the partial order formula_171 i.e.,
Proof. This follows from Chinese Remainder Theorem. 
Proof. We will use the universal property of the tensor product. Define a formula_175-bilinear function 
This is well-defined because for a given formula_177 with formula_178 co-prime there are only finitely many primes dividing formula_179 Let formula_180 be another formula_175-module with a formula_175-bilinear map formula_183 We have to show formula_184 factors through formula_185 uniquely, i.e., there exists a unique formula_175-linear map formula_187 such that formula_188 We define formula_189 as follows: for a given formula_190 there exist formula_191 and formula_192 such that formula_193 for all formula_194 Define formula_195 One can show formula_189 is well-defined, formula_175-linear, satisfies formula_198 and is unique with these properties. 
Proof. formula_201 
Remark. Using formula_203 where there are formula_204 summands, we give the right side the product topology and transport this topology via the isomorphism onto formula_205
The adele ring of a finite extension.
If formula_83 be a finite extension then formula_86 is a global field and thus formula_208 is defined and formula_209 We claim formula_122 can be identified with a subgroup of formula_211 Map formula_212 to formula_213 where formula_214 for formula_215 Then formula_216 is in the subgroup formula_217 if formula_218 for formula_219 and formula_220 for all formula_221 lying above the same place formula_56 of formula_78
With the help of this isomorphism, the inclusion formula_226 is given by
Furthermore, the principal adeles in formula_122 can be identified with a subgroup of principal adeles in formula_208 via the map
Proof. Let formula_231 be a basis of formula_86 over formula_78 Then for almost all formula_90
Furthermore, there are the following isomorphisms:
For the second we used the map: 
in which formula_238 is the canonical embedding and formula_239 We take on both sides the restricted product with respect to formula_240
The set of principal adeles in formula_208 is identified with the set formula_245 where the left side has formula_243 summands and we consider formula_3 as a subset of formula_131
The adele ring of vector-spaces and algebras.
Remark. If formula_254 is another finite set of places of formula_3 containing formula_256 then formula_252 is an open subring of formula_258
Now, we are able to give an alternative characterization of the adele ring. The adele ring is the union of all sets formula_252:
Equivalently formula_122 is the set of all formula_262 so that formula_263 for almost all formula_264 The topology of formula_122 is induced by the requirement that all formula_252 be open subrings of formula_131 Thus, formula_122 is a locally compact topological ring.
Fix a place formula_56 of formula_78 Let formula_256 be a finite set of places of formula_104 containing formula_56 and formula_274 Define
Then:
Furthermore, define
where formula_256 runs through all finite sets containing formula_279 Then:
via the map formula_281 The entire procedure above holds with a finite subset formula_282 instead of formula_283
By construction of formula_284 there is a natural embedding: formula_285 Furthermore, there exists a natural projection formula_286
The adele ring of a vector-space.
Let formula_110 be a finite dimensional vector-space over formula_3 and formula_289 a basis for formula_110 over formula_78 For each place formula_56 of formula_104 we write: 
We define the adele ring of formula_110 as
This definition is based on the alternative description of the adele ring as a tensor product equipped with the same topology we defined when giving an alternate definition of adele ring for number fields. We equip formula_297 with the restricted product topology. Then formula_298 and we can embed formula_110 in formula_297 naturally via the map formula_301
We give an alternative definition of the topology on formula_302 Consider all linear maps: formula_303 Using the natural embeddings formula_304 and formula_305 extend these linear maps to: formula_306 The topology on formula_297 is the coarsest topology for which all these extensions are continuous.
We can define the topology in a different way. Fixing a basis for formula_110 over formula_3 results in an isomorphism formula_310 Therefore fixing a basis induces an isomorphism formula_311 We supply the left hand side with the product topology and transport this topology with the isomorphism onto the right hand side. The topology doesn't depend on the choice of the basis, because another basis defines a second isomorphism. By composing both isomorphisms, we obtain a linear homeomorphism which transfers the two topologies into each other. More formally
where the sums have formula_313 summands. In case of formula_314 the definition above is consistent with the results about the adele ring of a finite extension formula_315
The adele ring of an algebra.
Let formula_316 be a finite-dimensional algebra over formula_78 In particular, formula_316 is a finite-dimensional vector-space over formula_78 As a consequence, formula_320 is defined and formula_321 Since we have a multiplication on formula_122 and formula_323 we can define a multiplication on formula_324 via:
As a consequence, formula_320 is an algebra with a unit over formula_131 Let formula_328 be a finite subset of formula_323 containing a basis for formula_316 over formula_78 For any finite place formula_56 we define formula_333 as the formula_62-module generated by formula_328 in formula_336 For each finite set of places, formula_337 we define
One can show there is a finite set formula_339 so that formula_340 is an open subring of formula_341 if formula_342 Furthermore formula_320 is the union of all these subrings and for formula_344 the definition above is consistent with the definition of the adele ring.
Trace and norm on the adele ring.
Let formula_83 be a finite extension. Since formula_346 and formula_347 from Lemma above we can interpret formula_122 as a closed subring of formula_211 We write formula_350 for this embedding. Explicitly for all places formula_85 of formula_86 above formula_56 and for any formula_354
Let formula_355 be a tower of global fields. Then:
Furthermore, restricted to the principal adeles formula_357 is the natural injection formula_358
Let formula_289 be a basis of the field extension formula_315 Then each formula_361 can be written as formula_362 where formula_363 are unique. The map formula_364 is continuous. We define formula_365 depending on formula_366 via the equations:
Now, we define the trace and norm of formula_368 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:
Furthermore, for formula_372formula_373 and formula_374 are identical to the trace and norm of the field extension formula_315 For a tower of fields formula_376 we have:
Moreover, it can be proven that:
Properties of the adele ring.
Remark. The result above also holds for the adele ring of vector-spaces and algebras over formula_78
Proof. We prove the case formula_385 To show formula_386 is discrete it is sufficient to show the existence of a neighbourhood of formula_387 which contains no other rational number. The general case follows via translation. Define
formula_389 is an open neighbourhood of formula_390 We claim formula_391 Let formula_392 then formula_393 and formula_394 for all formula_35 and therefore formula_396 Additionally, we have formula_397 and therefore formula_398 Next, we show compactness, define:
We show each element in formula_400 has a representative in formula_401 that is for each formula_402 there exists formula_393 such that formula_404 Let formula_405 be arbitrary and formula_35 be a prime for which formula_407 Then there exists formula_408 with formula_409 and formula_410 Replace formula_368 with formula_412 and let formula_413 be another prime. Then: 
The other equations follow similarly.
Characterisation of the idele group.
Proof. The class number of a number field is finite so let formula_419 be the ideals, representing the classes in formula_420 These ideals are generated by a finite number of prime ideals formula_421 Let formula_134 be a finite set of places containing formula_423 and the finite places corresponding to formula_421 Consider the isomorphism:
induced by 
At infinite places the statement is obvious so we prove the statement for finite places. The inclusion ″formula_427″ is obvious. Let formula_428 The corresponding ideal formula_429 belongs to a class formula_430 meaning formula_431 for a principal ideal formula_432 The idele formula_433 maps to the ideal formula_434 under the map formula_435 That means formula_436 Since the prime ideals in formula_434 are in formula_417 it follows formula_439 for all formula_440 that means formula_441 for all formula_442 It follows, that formula_443 therefore formula_444
Applications.
Finiteness of the class number of a number field.
In the previous section we used the fact that the class number of a number field is finite. Here we would like to prove this statement:
Proof. The map 
is surjective and therefore formula_448 is the continuous image of the compact set formula_449 Thus, formula_448 is compact. In addition it is discrete and so finite.
Remark. There is a similar result for the case of a global function field. In this case, the so-called divisor group is defined. It can be shown, that the quotient of the set of all divisors of degree formula_387 by the set of the principal divisors is a finite group.
Group of units and Dirichlet's unit theorem.
Let formula_452 be a finite set of places. Define
Then formula_454 is a subgroup of formula_455 containing all elements formula_456 satisfying formula_457 for all formula_458 Since formula_459 is discrete in formula_460 formula_454 is a discrete subgroup of formula_462 and with the same argument, formula_454 is discrete in formula_464
An alternative definition is: formula_465 where formula_466 is a subring of formula_3 defined by 
As a consequence, formula_466 contains all elements formula_470 which fulfil formula_471 for all formula_458
Proof. Define
formula_476 is compact and the set described above is the intersection of formula_476 with the discrete subgroup formula_478 in formula_479 and therefore finite. 
Proof. All roots of unity of formula_3 have absolute value formula_487 so formula_488 For converse note that Lemma 1 with formula_489 and any formula_256 implies formula_110 is finite. Moreover formula_492 for each finite set of places formula_493 Finally Suppose there exists formula_494 which is not a root of unity of formula_78 Then formula_496 for all formula_497 contradicting the finiteness of formula_498
Remark. The Unit Theorem is a generalisation of Dirichlet's Unit Theorem. To see this let formula_3 be a number field. We already know that formula_515 set formula_516 and note formula_517 Then we have:
Approximation theorems.
Remark. The global field is discrete in its adele ring. The strong approximation theorem tells us that, if we omit one place (or more), the property of discreteness of formula_3 is turned into a denseness of formula_78
Hasse principle.
Remark. This is the Hasse principle for quadratic forms. For polynomials of degree larger than 2 the Hasse principle isn't valid in general. The idea of the Hasse principle (also known as local–global principle) is to solve a given problem of a number field formula_3 by doing so in its completions formula_58 and then concluding on a solution in formula_78
Characters on the adele ring.
Definition. Let formula_1 be a locally compact abelian group. The character group of formula_1 is the set of all characters of formula_1 and is denoted by formula_547 Equivalently formula_548 is the set of all continuous group homomorphisms from formula_1 to formula_550 We equip formula_548 with the topology of uniform convergence on compact subsets of formula_552 One can show that formula_548 is also a locally compact abelian group.
Proof. By reduction to local coordinates it is sufficient to show each formula_58 is self-dual. This can done by using a fixed character of formula_540 We illustrate this idea by showing formula_116 is self-dual. Define: 
Then the following map is an isomorphism which respects topologies: 
Tate's thesis.
With the help of the characters of formula_217 we can do Fourier analysis on the adele ring. John Tate in his thesis "Fourier analysis in number fields and Heckes 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. We can define adelic forms of these functions and we can represent them as integrals over the adele ring or the idele group, with respect to corresponding Haar measures. We can show functional equations and meromorphic continuations of these functions. For example, for all formula_587 with formula_588
where formula_590 is the unique Haar measure on formula_591 normalized such that formula_592 has volume one and is extended by zero to the finite adele ring. As a result the Riemann zeta function can be written as an integral over (a subset of) the adele ring.
Automorphic forms.
The theory of automorphic forms is a generalization of Tate's thesis by replacing the idele group with analogous higher dimensional groups. To see this note:
Based on these identification a natural generalization would be to replace the idele group and the 1-idele with:
And finally
where formula_596 is the centre of formula_597 Then we define an automorphic form as an element of formula_598 In other words an automorphic form is a functions on formula_599 satisfying certain algebraic and analytic conditions. For studying automorphic forms, it is important to know the representations of the group formula_600 It is also possible to study automorphic L-functions, which can be described as integrals over formula_600
We could generalize even further by replacing formula_52 with a number field and formula_603 with an arbitrary reductive algebraic group.
Further applications.
A generalisation of Artin reciprocity law leads to the connection of representations of formula_604 and of Galois representations of formula_3 (Langlands program).
The idele class group is a key object of class field theory, which describes abelian extensions of the field. The product of the local reciprocity maps in local class field theory gives a homomorphism of the idele group to the Galois group of the maximal abelian extension of the global field. The Artin reciprocity law, which is a high level generalisation of the Gauss quadratic reciprocity law, states that the product vanishes on the multiplicative group of the number field. Thus, we obtain the global reciprocity map of the idele class group to the abelian part of the absolute Galois group of the field.
The self-duality of the adele ring of the function field of a curve over a finite field easily implies the Riemann–Roch theorem and the duality theory for the curve.