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. An adele derives from a particular kind of idele. "Idele" derives from the French "idèle" and was coined by the French mathematician Claude Chevalley. The word stands for 'ideal element' (abbreviated: id.el.). Adele (French: "adèle") stands for 'additive idele' (that is, additive ideal element). The ring of adeles allows one to describe the Artin reciprocity law, which is a generalisation of quadratic reciprocity, and other reciprocity laws over finite fields. In addition, it is a classical theorem from Weil that G-bundles on an algebraic curve over a finite field can be described in terms of adeles for a reductive group G. Adeles are also connected with the adelic algebraic groups and adelic curves. The study of geometry of numbers over the ring of adeles of a number field is called adelic geometry.

Definition

Let K be a global field (a finite extension of \mathbf{Q} or the function field of a curve over a finite field). The adele ring of K is the subring consisting of the tuples (a_\nu) where a_\nu lies in the subring for all but finitely many places \nu. Here the index \nu ranges over all valuations of the global field K, K_\nu is the completion at that valuation and the corresponding valuation ring.

Motivation

The ring of adeles solves the technical problem of "doing analysis on the rational numbers \mathbf{Q}." The classical solution was to pass to the standard metric completion \mathbf{R} 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, as was classified by Ostrowski. The Euclidean absolute value, denoted, is only one among many others, |\cdot |_p, but the ring of adeles makes it possible to comprehend and. This has the advantage of enabling analytic techniques while also retaining information about the primes, since their structure is embedded by the restricted infinite product. The purpose of the adele ring is to look at all completions of K at once. The adele ring is defined with the restricted product, rather than the Cartesian product. There are two reasons for this:

Why the restricted product?

The restricted infinite product is a required technical condition for giving the number field \mathbf{Q} a lattice structure inside of, making it possible to build a theory of Fourier analysis (cf. Harmonic analysis) in the adelic setting. This is analogous to the situation in algebraic number theory where the ring of integers of an algebraic number field embeds""as 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 by constructing the ring of adeles as a tensor product of rings. If defining the ring of integral adeles as the ring""then the ring of adeles can be equivalently defined as The restricted product structure becomes transparent after looking at explicit elements in this ring. The image of an element inside of the unrestricted product is the element The factor ba_p/c lies in whenever p is not a prime factor of c, which is the case for all but finitely many primes p.

Origin of the name

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" (French: adèle) stands for additive idele. Thus, an adele is an additive ideal element.

Examples

Ring of adeles for the rational numbers

The rationals have a valuation for every prime number p, with, and one infinite valuation ∞ with. 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 of the projective line over a finite field. Its valuations correspond to points x of, i.e. maps over For instance, there are q+1 points of the form. In this case is the completed stalk of the structure sheaf at x (i.e. functions on a formal neighbourhood of x) and is its fraction field. Thus The same holds for any smooth proper curve over a finite field, the restricted product being over all points of x \in X.

Related notions

The group of units in the adele ring is called the idele group The quotient of the ideles by the subgroup 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 K^{ab} is the maximal abelian algebraic extension of K and means the profinite completion of the group.

Giving adelic formulation of Picard group of a curve

If is a smooth proper curve then its Picard group is and its divisor group is. Similarly, if G is a semisimple algebraic group (e.g. SL_n, it also holds for GL_n) then Weil uniformisation says that Applying this to gives the result on the Picard group.

Tate's thesis

There is a topology on for which the quotient is compact, allowing one to do harmonic analysis on it. John Tate in his thesis "Fourier analysis in number fields and Hecke 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 exactly as the finite fields case. John Tate proved that Serre duality on X can be deduced by working with this adele ring. Here L is a line bundle on X.

Notation and basic definitions

Global fields

Throughout this article, K is a global field, meaning it is either a number field (a finite extension of \Q) or a global function field (a finite extension of for p prime and r \in \N). By definition a finite extension of a global field is itself a global field.

Valuations

For a valuation v of K it can be written K_v for the completion of K with respect to v. If v is discrete it can be written O_v for the valuation ring of K_v and for the maximal ideal of O_v. If this is a principal ideal denoting the uniformising element by \pi_v. A non-Archimedean valuation is written as v<\infty or and an Archimedean valuation as v | \infty. Then assume all valuations to be non-trivial. There is a one-to-one identification of valuations and absolute values. Fix a constant C>1, the valuation v is assigned the absolute value |\cdot|v, defined as: Conversely, the absolute value |\cdot| is assigned the valuation defined as: A place of K is a representative of an equivalence class of valuations (or absolute values) of K. Places corresponding to non-Archimedean valuations are called finite, whereas places corresponding to Archimedean valuations are called infinite. Infinite places of a global field form a finite set, which is denoted by P{\infty}. Define and let be its group of units. Then

Finite extensions

Let L/K be a finite extension of the global field K. Let w be a place of L and v a place of K. If the absolute value |\cdot|_w restricted to K is in the equivalence class of v, then w lies above v, which is denoted by w | v, and defined as: (Note that both products are finite.) If w|v, K_v can be embedded in L_w. Therefore, K_v is embedded diagonally in L_v. With this embedding L_v is a commutative algebra over K_v with degree

The adele ring

The set of finite adeles of a global field K, denoted is defined as the restricted product of K_v with respect to the O_v: It is equipped with the restricted product topology, the topology generated by restricted open rectangles, which have the following form: where E is a finite set of (finite) places and are open. With component-wise addition and multiplication is also a ring. The adele ring of a global field K is defined as the product of with the product of the completions of K at its infinite places. The number of infinite places is finite and the completions are either \R or \C. 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 K. In the following, it is written as 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 a \in K, then for almost all v. This shows the map is well-defined. It is also injective because the embedding of K in K_v is injective for all v. Remark. By identifying K with its image under the diagonal map it is regarded as a subring of The elements of K are called the principal adeles of Definition. Let S be a set of places of K. Define the set of the S-adeles of K as Furthermore, if the result is:

The adele ring of rationals

By Ostrowski's theorem the places of \Q are it is possible to identify a prime p with the equivalence class of the p-adic absolute value and \infty with the equivalence class of the absolute value defined as: The completion of \Q with respect to the place p is \Q_p with valuation ring \Z_p. For the place \infty the completion is \R. Thus: Or for short the difference between restricted and unrestricted product topology can be illustrated using a sequence in : 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 and for each restricted open rectangle it has: for and therefore for all p \notin F. As a result for almost all n \in \N. In this consideration, E and F are finite subsets of the set of all places.

Alternative definition for number fields

Definition (profinite integers). The profinite integers are defined as the profinite completion of the rings \Z /n\Z with the partial order i.e., Proof. This follows from the Chinese Remainder Theorem. Proof. Use the universal property of the tensor product. Define a \Z-bilinear function This is well-defined because for a given with m,n co-prime there are only finitely many primes dividing n. Let M be another \Z-module with a \Z-bilinear map It must be the case that \Phi factors through \Psi uniquely, i.e., there exists a unique \Z-linear map such that can be defined as follows: for a given (u_p)_p there exist u \in \N and such that for all p. Define One can show is well-defined, \Z-linear, satisfies and is unique with these properties. Proof. Remark. Using where there are [K:\Q] summands, give the right side receives the product topology and transport this topology via the isomorphism onto

The adele ring of a finite extension

If L/K be a finite extension, then L is a global field. Thus is defined, and can be identified with a subgroup of Map to where for w|v. Then is in the subgroup if a_w \in K_v for w | v and a_w=a_{w'} for all w, w' lying above the same place v of K. With the help of this isomorphism, the inclusion is given by Furthermore, the principal adeles in can be identified with a subgroup of principal adeles in via the map Proof. Let be a basis of L over K. Then for almost all v, Furthermore, there are the following isomorphisms: For the second use the map: in which is the canonical embedding and w | v. The restricted product is taken on both sides with respect to The set of principal adeles in is identified with the set where the left side has [L:K] summands and K is considered as a subset of

The adele ring of vector-spaces and algebras

Remark. If P' is another finite set of places of K containing P then is an open subring of Now, an alternative characterisation of the adele ring can be presented. The adele ring is the union of all sets : Equivalently is the set of all x=(x_v)v so that for almost all v < \infty. The topology of is induced by the requirement that all be open subrings of Thus, is a locally compact topological ring. Fix a place v of K. Let P be a finite set of places of K, containing v and P\infty. Define Then: Furthermore, define where P runs through all finite sets containing Then: via the map The entire procedure above holds with a finite subset instead of {v}. By construction of there is a natural embedding: Furthermore, there exists a natural projection

The adele ring of a vector-space

Let E be a finite dimensional vector-space over K and a basis for E over K. For each place v of K: The adele ring of E is defined as This definition is based on the alternative description of the adele ring as a tensor product equipped with the same topology that was defined when giving an alternate definition of adele ring for number fields. Next, is equipped with the restricted product topology. Then and E is embedded in naturally via the map An alternative definition of the topology on can be provided. Consider all linear maps: E \to K. Using the natural embeddings and extend these linear maps to: The topology on is the coarsest topology for which all these extensions are continuous. The topology can be defined in a different way. Fixing a basis for E over K results in an isomorphism Therefore fixing a basis induces an isomorphism The left-hand side is supplied 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, a linear homeomorphism which transfers the two topologies into each other is obtained. More formally where the sums have n summands. In case of E=L, the definition above is consistent with the results about the adele ring of a finite extension L/K.

The adele ring of an algebra

Let A be a finite-dimensional algebra over K. In particular, A is a finite-dimensional vector-space over K. As a consequence, is defined and Since there is multiplication on and A, a multiplication on can be defined via: As a consequence, is an algebra with a unit over Let \mathcal{B} be a finite subset of A, containing a basis for A over K. For any finite place v, M_v is defined as the O_v-module generated by \mathcal{B} in A_v. For each finite set of places, define One can show there is a finite set P_0, so that is an open subring of if Furthermore is the union of all these subrings and for A=K, the definition above is consistent with the definition of the adele ring.

Trace and norm on the adele ring

Let L/K be a finite extension. Since and from the Lemma above, can be interpreted as a closed subring of For this embedding, write. Explicitly for all places w of L above v and for any Let M/L/K be a tower of global fields. Then: Furthermore, restricted to the principal adeles is the natural injection K \to L. Let be a basis of the field extension L/K. Then each can be written as where are unique. The map is continuous. Define \alpha_{ij} depending on \alpha via the equations: Now, define the trace and norm of \alpha 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 and are identical to the trace and norm of the field extension L/K. For a tower of fields M/L/K, the result is: 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 K. Proof. Prove the case K=\Q. To show is discrete it is sufficient to show the existence of a neighbourhood of 0 which contains no other rational number. The general case follows via translation. Define U is an open neighbourhood of It is claimed that Let then and for all p and therefore Additionally, and therefore \beta=0. Next, to show compactness, define: Each element in has a representative in W, that is for each there exists such that Let be arbitrary and p be a prime for which Then there exists with and Replace \alpha with \alpha-r_p and let q \neq p be another prime. Then: Next, it can be claimed that: The reverse implication is trivially true. The implication is true, because the two terms of the strong triangle inequality are equal if the absolute values of both integers are different. As a consequence, the (finite) set of primes for which the components of \alpha are not in \Z_p is reduced by 1. With iteration, it can be deduced that there exists r\in \Q such that Now select s \in \Z such that Then The continuous projection is surjective, therefore as the continuous image of a compact set, is compact. Proof. The first two equations can be proved in an elementary way. By definition is divisible if for any n \in \N and the equation nx=y has a solution It is sufficient to show is divisible but this is true since is a field with positive characteristic in each coordinate. For the last statement note that because the finite number of denominators in the coordinates of the elements of can be reached through an element q \in \Q. As a consequence, it is sufficient to show is dense, that is each open subset contains an element of \Z. Without loss of generality, it can be assumed that because is a neighbourhood system of 0 in \Z_p. By Chinese Remainder Theorem there exists l \in \Z such that Since powers of distinct primes are coprime, l \in V follows. Remark. is not uniquely divisible. Let and n \geq 2 be given. Then both satisfy the equation nx=y and clearly (x_2 is well-defined, because only finitely many primes divide n). In this case, being uniquely divisible is equivalent to being torsion-free, which is not true for since nx_2 = 0, but x_2 \neq 0 and n \neq 0. Remark. The fourth statement is a special case of the strong approximation theorem.

Haar measure on the adele ring

Definition. A function is called simple if where are measurable and for almost all v.

The idele group

Definition. Define the idele group of K as the group of units of the adele ring of K, that is The elements of the idele group are called the ideles of K. Remark. I_K is equipped with a topology so that it becomes a topological group. The subset topology inherited from is not a suitable candidate since the group of units of a topological ring equipped with subset topology may not be a topological group. For example, the inverse map in is not continuous. The sequence converges to To see this let U be neighbourhood of 0, without loss of generality it can be assumed: Since for all p, x_n-1 \in U for n large enough. However, as was seen above the inverse of this sequence does not converge in Proof. Since R is a topological ring, it is sufficient to show that the inverse map is continuous. Let be open, then is open. It is necessary to show is open or equivalently, that is open. But this is the condition above. The idele group is equipped with the topology defined in the Lemma making it a topological group. Definition. For S a subset of places of K set: Proof. Prove the identity for I_K; the other two follow similarly. First show the two sets are equal: In going from line 2 to 3, x as well as x^{-1}=y have to be in meaning x_v \in O_v for almost all v and for almost all v. Therefore, for almost all v. Now, it is possible to show the topology on the left-hand side equals the topology on the right-hand side. Obviously, every open restricted rectangle is open in the topology of the idele group. On the other hand, for a given which is open in the topology of the idele group, meaning is open, so for each u \in U there exists an open restricted rectangle, which is a subset of U and contains u. Therefore, U is the union of all these restricted open rectangles and therefore is open in the restricted product topology. Proof. The local compactness follows from the description of I_{K,S} as a restricted product. It being a topological group follows from the above discussion on the group of units of a topological ring. A neighbourhood system of is a neighbourhood system of 1 \in I_K. Alternatively, take all sets of the form: where U_v is a neighbourhood of and for almost all v. Since the idele group is a locally compact, there exists a Haar measure d^\times x on it. This can be normalised, so that This is the normalisation used for the finite places. In this equation, is the finite idele group, meaning the group of units of the finite adele ring. For the infinite places, use the multiplicative lebesgue measure

The idele group of a finite extension

Proof. Map to with the property for w|v. Therefore, I_K can be seen as a subgroup of I_L. An element is in this subgroup if and only if his components satisfy the following properties: for w | v and a_w=a_{w'} for w|v and w' | v for the same place v of K.

The case of vector spaces and algebras

The idele group of an algebra

Let A be a finite-dimensional algebra over K. Since is not a topological group with the subset-topology in general, equip with the topology similar to I_K above and call the idele group. The elements of the idele group are called idele of A.

Alternative characterisation of the idele group

Norm on the idele group

The trace and the norm should be transfer from the adele ring to the idele group. It turns out the trace can't be transferred so easily. However, it is possible to transfer the norm from the adele ring to the idele group. Let Then and therefore, it can be said that in injective group homomorphism Since it is invertible, is invertible too, because Therefore As a consequence, the restriction of the norm-function introduces a continuous function:

The Idele class group

Proof. Since K^{\times} is a subset of for all v, the embedding is well-defined and injective. Defenition. In analogy to the ideal class group, the elements of K^{\times} in I_K are called principal ideles of I_K. The quotient group is called idele class group of K. This group is related to the ideal class group and is a central object in class field theory. Remark. K^\times is closed in I_K, therefore C_K is a locally compact topological group and a Hausdorff space.

Properties of the idele group

Absolute value on the idele group of K and 1-idele

Definition. For define: Since \alpha is an idele this product is finite and therefore well-defined. Remark. The definition can be extended to by allowing infinite products. However, these infinite products vanish and so |\cdot| vanishes on |\cdot| will be used to denote both the function on I_K and Proof. Let where it is used that all products are finite. The map is continuous which can be seen using an argument dealing with sequences. This reduces the problem to whether |\cdot| is continuous on K_v. However, this is clear, because of the reverse triangle inequality. Definition. The set of 1-idele can be defined as: I_K^1 is a subgroup of I_K. Since it is a closed subset of Finally the -topology on I_K^1 equals the subset-topology of I_K on I_K^1. Proof. Proof of the formula for number fields, the case of global function fields can be proved similarly. Let K be a number field and It has to be shown that: For finite place v for which the corresponding prime ideal does not divide (a), v(a)=0 and therefore |a|v=1. This is valid for almost all There is: In going from line 1 to line 2, the identity was used where v is a place of K and w is a place of L, lying above v. Going from line 2 to line 3, a property of the norm is used. The norm is in \Q so without loss of generality it can be assumed that a \in \Q. Then a possesses a unique integer factorisation: where v_p \in \Z is 0 for almost all p. By Ostrowski's theorem all absolute values on \Q are equivalent to the real absolute value or a p-adic absolute value. Therefore: Proof. Let C be the constant from the lemma. Let \pi_v be a uniformising element of O_v. Define the adele via with k_v \in \Z minimal, so that for all v \neq v_0. Then k_v=0 for almost all v. Define with so that This works, because k_v=0 for almost all v. By the Lemma there exists so that for all v \neq v_0. Proof. Since K^\times is discrete in I_K it is also discrete in I_K^1. To prove the compactness of let C is the constant of the Lemma and suppose satisfying is given. Define: Clearly W\alpha is compact. It can be claimed that the natural projection is surjective. Let be arbitrary, then: and therefore It follows that By the Lemma there exists such that for all v, and therefore proving the surjectivity of the natural projection. Since it is also continuous the compactness follows. Proof. Consider the map This map is well-defined, since |a_p|_p=1 for all p and therefore Obviously \phi is a continuous group homomorphism. Now suppose Then there exists such that By considering the infinite place it can be seen that q=1 proves injectivity. To show surjectivity let The absolute value of this element is 1 and therefore Hence and there is: Since It has been concluded that \phi is surjective. Proof. The isomorphisms are given by:

Relation between ideal class group and idele class group

Proof. Let v be a finite place of K and let |\cdot|_v be a representative of the equivalence class v. Define Then is a prime ideal in O. The map is a bijection between finite places of K and non-zero prime ideals of O. The inverse is given as follows: a prime ideal is mapped to the valuation given by The following map is well-defined: The map (\cdot) is obviously a surjective homomorphism and The first isomorphism follows from fundamental theorem on homomorphism. Now, both sides are divided by K^{\times}. This is possible, because Please, note the abuse of notation: On the left hand side in line 1 of this chain of equations, (\cdot) stands for the map defined above. Later, the embedding of K^{\times} into is used. In line 2, the definition of the map is used. Finally, use that O is a Dedekind domain and therefore each ideal can be written as a product of prime ideals. In other words, the map (\cdot) is a K^\times-equivariant group homomorphism. As a consequence, the map above induces a surjective homomorphism To prove the second isomorphism, it has to be shown that Consider Then because v(\xi_v)=0 for all v. On the other hand, consider with which allows to write As a consequence, there exists a representative, such that: Consequently, and therefore The second isomorphism of the theorem has been proven. For the last isomorphism note that \phi induces a surjective group homomorphism with Remark. Consider with the idele topology and equip J_K, with the discrete topology. Since is open for each is continuous. It stands, that is open, where so that

Decomposition of the idele group and idele class group of K

Proof. For each place v of so that for all |x|v belongs to the subgroup of \R+, generated by p. Therefore for each z \in I_K, |z| is in the subgroup of \R_+, generated by p. Therefore the image of the homomorphism is a discrete subgroup of \R_+, generated by p. Since this group is non-trivial, it is generated by Q=p^m for some m \in \N. Choose so that |z_1|=Q, then I_K is the direct product of I_K^1 and the subgroup generated by z_1. This subgroup is discrete and isomorphic to \Z. For define: The map is an isomorphism of \R_+ in a closed subgroup M of I_K and The isomorphism is given by multiplication: Obviously, \phi is a homomorphism. To show it is injective, let Since \alpha_v=1 for v < \infty, it stands that \beta_v=1 for v < \infty. Moreover, it exists a so that for v | \infty. Therefore, for v | \infty. Moreover implies where n is the number of infinite places of K. As a consequence \lambda=1 and therefore \phi is injective. To show surjectivity, let It is defined that and furthermore, \alpha_v=1 for v < \infty and for v | \infty. Define It stands, that Therefore, \phi is surjective. The other equations follow similarly.

Characterisation of the idele group

Proof. The class number of a number field is finite so let be the ideals, representing the classes in These ideals are generated by a finite number of prime ideals Let S be a finite set of places containing P_\infty and the finite places corresponding to Consider the isomorphism: induced by At infinite places the statement is immediate, so the statement has been proved for finite places. The inclusion ″\supset″ is obvious. Let The corresponding ideal belongs to a class meaning for a principal ideal (a). The idele maps to the ideal under the map That means Since the prime ideals in are in S, it follows for all v \notin S, that means for all v \notin S. It follows, that therefore

Applications

Finiteness of the class number of a number field

In the previous section the fact that the class number of a number field is finite had been used. Here this statement can be proved: Proof. The map is surjective and therefore is the continuous image of the compact set Thus, 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 0 by the set of the principal divisors is a finite group.

Group of units and Dirichlet's unit theorem

Let be a finite set of places. Define Then E(P) is a subgroup of K^{\times}, containing all elements satisfying v(\xi)=0 for all v \notin P. Since K^{\times} is discrete in I_K, E(P) is a discrete subgroup of \Omega(P) and with the same argument, E(P) is discrete in An alternative definition is: where K(P) is a subring of K defined by As a consequence, K(P) contains all elements \xi \in K, which fulfil for all v \notin P. Proof. Define W is compact and the set described above is the intersection of W with the discrete subgroup K^\times in I_K and therefore finite. Proof. All roots of unity of K have absolute value 1 so For converse note that Lemma 1 with c=C=1 and any P implies E is finite. Moreover for each finite set of places Finally suppose there exists \xi \in E, which is not a root of unity of K. Then for all n \in \N contradicting the finiteness of E. Remark. The Unit Theorem generalises Dirichlet's Unit Theorem. To see this, let K be a number field. It is already known that E=\mu(K), set P=P_\infty and note Then there is:

Approximation theorems

Remark. The global field is discrete in its adele ring. The strong approximation theorem tells us that, if one place (or more) is omitted, the property of discreteness of K is turned into a denseness of K.

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 K by doing so in its completions K_v and then concluding on a solution in K.

Characters on the adele ring

Definition. Let G be a locally compact abelian group. The character group of G is the set of all characters of G and is denoted by Equivalently \widehat{G} is the set of all continuous group homomorphisms from G to Equip \widehat{G} with the topology of uniform convergence on compact subsets of G. One can show that \widehat{G} is also a locally compact abelian group. Proof. By reduction to local coordinates, it is sufficient to show each K_v is self-dual. This can be done by using a fixed character of K_v. The idea has been illustrated by showing \R is self-dual. Define: Then the following map is an isomorphism which respects topologies:

Tate's thesis

With the help of the characters of Fourier analysis can be done on the adele ring. John Tate in his thesis "Fourier analysis in Number Fields and Hecke 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. Adelic forms of these functions can be defined and represented as integrals over the adele ring or the idele group, with respect to corresponding Haar measures. Functional equations and meromorphic continuations of these functions can be shown. For example, for all s \in \C with \Re(s) > 1, where d^\times x is the unique Haar measure on normalised such that 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 generalisation of Tate's thesis by replacing the idele group with analogous higher dimensional groups. To see this note: Based on these identification a natural generalisation would be to replace the idele group and the 1-idele with: And finally where Z_\R is the centre of Then it define an automorphic form as an element of In other words an automorphic form is a function on satisfying certain algebraic and analytic conditions. For studying automorphic forms, it is important to know the representations of the group It is also possible to study automorphic L-functions, which can be described as integrals over Generalise even further is possible by replacing \Q with a number field and with an arbitrary reductive algebraic group.

Further applications

A generalisation of Artin reciprocity law leads to the connection of representations of and of Galois representations of K (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 sweeping generalisation of the Gauss quadratic reciprocity law, states that the product vanishes on the multiplicative group of the number field. Thus, the global reciprocity map of the idele class group to the abelian part of the absolute Galois group of the field will be obtained. 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.

Sources