Adele ring

Let K be a global field (a finite extension of \mathbf{Q} or the function field of a curve X/\mathbf{F_{\mathit{q}}} over a finite field). The adele ring of K is the subring :\mathbf{A}_K\ = \ \prod (K_\nu,\mathcal{O}_\nu) \ \subseteq \ \prod K_\nu consisting of the tuples (a_\nu) where a_\nu lies in the subring \mathcal{O}_\nu \subset K_\nu 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 \mathcal{O}_\nu 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 p \in \mathbf{Z}, as classified by Ostrowski's theorem. The Euclidean absolute value, denoted |\cdot|_\infty, 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: • For each element of K the valuations are zero for almost all places, i.e., for all places except a finite number. So, the global field can be embedded in the restricted product. • The restricted product is a locally compact space, while the Cartesian product is not. Therefore, there cannot be any application of harmonic analysis to the Cartesian product. This is because local compactness ensures the existence (and uniqueness) of Haar measure, a crucial tool in analysis on groups in general. 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 \mathbf{A}_\mathbf{Q}, 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\mathcal{O}_K \hookrightarrow Kas 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 \mathbf{A}_\mathbf{Z} as the ring\mathbf{A}_\mathbf{Z} = \mathbf{R}\times\hat{\mathbf{Z}} = \mathbf{R}\times \prod_p \mathbf{Z}_p,then the ring of adeles can be equivalently defined as\begin{align} \mathbf{A}_\mathbf{Q} &= \mathbf{Q}\otimes_\mathbf{Z}\mathbf{A}_\mathbf{Z} \\ &= \mathbf{Q}\otimes_\mathbf{Z} \left( \mathbf{R}\times \prod_{p} \mathbf{Z}_p \right). \end{align}The restricted product structure becomes transparent after looking at explicit elements in this ring. The image of an element b/c\otimes(r,(a_p)) \in \mathbf{A}_\mathbf{Q} inside of the unrestricted product \mathbf{R}\times \prod_p \mathbf{Q}_p is the element \left(\frac{br\vphantom{ba_p}}{c}, \left(\frac{ba_p}{c}\right) \right). The factor ba_p/c lies in \mathbf{Z}_p 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: ) stands for additive idele. Thus, an adele is an additive ideal element. ==Examples==

Ring of adeles for the rational numbers The rationals K = \bold{Q} have a valuation for every prime number p, with ( K_\nu,\mathcal{O}_\nu)=(\mathbf{Q}_p,\mathbf{Z}_p), and one infinite valuation ∞ with \mathbf{Q}_\infty = \mathbf{R}. Thus an element of :\mathbf{A}_\mathbf{Q}\ = \ \mathbf{R}\times \prod_p (\mathbf{Q}_p,\mathbf{Z}_p) 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=\mathbf{F}_q(\mathbf{P}^1)=\mathbf{F}_q(t) of the projective line over a finite field. Its valuations correspond to points x of X=\mathbf{P}^1, i.e. maps over \text{Spec}\mathbf{F}_{q} :x\ :\ \text{Spec}\mathbf{F}_{q^n}\ \longrightarrow \ \mathbf{P}^1. For instance, there are q+1 points of the form \text{Spec}\mathbf{F}_{q}\ \longrightarrow \ \mathbf{P}^1. In this case \mathcal{O}_\nu=\widehat{\mathcal{O}}_{X,x} is the completed stalk of the structure sheaf at x (i.e. functions on a formal neighbourhood of x) and K_\nu=K_{X,x} is its fraction field. Thus :\mathbf{A}_{\mathbf{F}_q(\mathbf{P}^1)}\ =\ \prod_{x\in X} (\mathcal{K}_{X,x},\widehat{\mathcal{O}}_{X,x}). The same holds for any smooth proper curve X/\mathbf{F_{\mathit{q}}} 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 :I_K = \mathbf{A}_{K}^\times. The quotient of the ideles by the subgroup K^\times \subseteq I_K is called the idele class group :C_K\ =\ I_K/K^\times. The integral adeles are the subring :\mathbf{O}_K\ =\ \prod O_\nu \ \subseteq \ \mathbf{A}_K. ==Applications==

Stating Artin reciprocity The Artin reciprocity law says that for a global field K, :\widehat{C_K} = \widehat{\mathbf{A}_K^\times/K^\times} \ \simeq \ \text{Gal}(K^\text{ab}/K) where K^{ab} is the maximal abelian algebraic extension of K and \widehat{(\dots)} means the profinite completion of the group. Giving adelic formulation of Picard group of a curve If X/\mathbf{F_{\mathit{q}}} is a smooth proper curve then its Picard group is :\text{Pic}(X) \ = \ K^\times\backslash \mathbf{A}^\times_X/\mathbf{O}_X^\times and its divisor group is \text{Div}(X)=\mathbf{A}^\times_X/\mathbf{O}_X^\times. Similarly, if G is a semisimple algebraic group (e.g. SL_n, it also holds for GL_n) then Weil uniformisation says that :\text{Bun}_G(X) \ = \ G(K)\backslash G(\mathbf{A}_X)/G(\mathbf{O}_X). Applying this to G=\mathbf{G}_m gives the result on the Picard group. Tate's thesis There is a topology on \mathbf{A}_K for which the quotient \mathbf{A}_K/K 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 \mathbf{C}(X) exactly as the finite fields case. John Tate proved that Serre duality on X :H^1(X,\mathcal{L})\ \simeq \ H^0(X,\Omega_X\otimes\mathcal{L}^{-1})^* can be deduced by working with this adele ring \mathbf{A}_{\mathbf{C}(X)}. 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 \mathbb{F}_{p^r}(t) 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 \mathfrak{m}_v 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 or v \nmid \infty 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: :\forall x \in K: \quad |x|_v := \begin{cases} C^{-v(x)} & x \neq 0 \\ 0 & x=0 \end{cases} Conversely, the absolute value |\cdot| is assigned the valuation v_, defined as: :\forall x \in K^\times: \quad v_(x):= - \log_C(|x|). 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 \textstyle \widehat{O}:= \prod_{v and let \widehat{O}^{\times} be its group of units. Then \textstyle \widehat{O}^{\times}=\prod_{v 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: :\begin{align} L_v&:=\prod_{w | v} L_w,\\ \widetilde{O_v} &:=\prod_{w | v}O_w. \end{align} (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 :\sum_{w|v}[L_w:K_v]=[L:K]. ==The adele ring==

The set of finite adeles of a global field K, denoted \mathbb{A}_{K,\text{fin}}, is defined as the restricted product of K_v with respect to the O_v: :\mathbb{A}_{K,\text{fin}}:= {\prod_{v It is equipped with the restricted product topology, the topology generated by restricted open rectangles, which have the following form: :U=\prod_{v \in E} U_v \times \prod_{v \notin E} O_v \subset {\prod_{v where E is a finite set of (finite) places and U_v \subset K_v are open. With component-wise addition and multiplication \mathbb{A}_{K,\text{fin}} is also a ring. The adele ring of a global field K is defined as the product of \mathbb{A}_{K,\text{fin}} 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: :\mathbb{A}_K:=\mathbb{A}_{K,\text{fin}}\times \prod_{v | \infty} K_v= {\prod_{v 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 :\mathbb{A}_K= {\prod_v}^' K_v, 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. :Lemma. There is a natural embedding of K into \mathbb{A}_K given by the diagonal map: a \mapsto (a,a,\ldots). Proof. If a \in K, then a \in O_v^{\times} 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 \mathbb{A}_K. The elements of K are called the principal adeles of \mathbb{A}_K. Definition. Let S be a set of places of K. Define the set of the S-adeles of K as : \mathbb{A}_{K,S} := {\prod_{v \in S}}^' K_v. Furthermore, if :\mathbb{A}_K^S := {\prod_{v \notin S}}^' K_v the result is: \mathbb{A}_K=\mathbb{A}_{K,S} \times \mathbb{A}_K^S. The adele ring of rationals By Ostrowski's theorem the places of \Q are \{p \in \N :p \text{ prime}\} \cup \{\infty\}, 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 |\cdot|_\infty defined as: :\forall x \in \Q: \quad |x|_\infty:= \begin{cases} x & x \geq 0 \\ -x & x 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: :\begin{align} \mathbb{A}_{\Q,\text{fin}} &= {\prod_{p Or for short : \mathbb{A}_{\Q} = {\prod_{p \leq \infty}}^' \Q_p,\qquad \Q_\infty:=\R. the difference between restricted and unrestricted product topology can be illustrated using a sequence in \mathbb{A}_\Q: :Lemma. Consider the following sequence in \mathbb{A}_\Q: ::\begin{align} x_1&=\left(\frac 1 2 ,1,1,\ldots\right)\\ x_2&=\left(1,\frac 1 3 ,1,\ldots\right)\\ x_3&=\left(1,1,\frac 1 5 ,1,\ldots\right)\\ x_4&=\left(1,1,1,\frac 1 7 ,1,\ldots\right)\\ & \vdots \end{align} :In the product topology this converges to (1,1,\ldots), but it does not converge at all in the restricted product topology. 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 a=(a_p)_p \in \mathbb{A}_{\Q} and for each restricted open rectangle \textstyle U=\prod_{p \in E}U_p \times \prod_{p \notin E}\Z_p, it has: \tfrac{1}{p}-a_p \notin \Z_p for a_p \in \Z_p and therefore \tfrac{1}{p}-a_p \notin \Z_p for all p \notin F. As a result x_n-a \notin U 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 n \geq m \Leftrightarrow m | n, i.e., :\widehat{\Z}:=\varprojlim_n \Z /n\Z, :Lemma. \textstyle \widehat{\Z} \cong \prod_p \Z_p. Proof. This follows from the Chinese Remainder Theorem. :Lemma. \mathbb{A}_{\Q, \text{fin}}= \widehat{\Z}\otimes_{\Z} \Q. Proof. Use the universal property of the tensor product. Define a \Z-bilinear function :\begin{cases} \Psi: \widehat{\Z}\times \Q \to \mathbb{A}_{\Q,\text{fin}} \\ \left ((a_p)_p,q \right ) \mapsto (a_pq)_p \end{cases} This is well-defined because for a given q = \tfrac{m}{n} \in \Q with m,n co-prime there are only finitely many primes dividing n. Let M be another \Z-module with a \Z-bilinear map \Phi: \widehat{\Z} \times \Q \to M. It must be the case that \Phi factors through \Psi uniquely, i.e., there exists a unique \Z-linear map \tilde{\Phi}: \mathbb{A}_{\Q,\text{fin}} \to M such that \Phi = \tilde{\Phi} \circ \Psi. \tilde{\Phi} can be defined as follows: for a given (u_p)_p there exist u \in \N and (v_p)_p \in \widehat{\Z} such that u_p=\tfrac{1}{u}\cdot v_p for all p. Define \tilde{\Phi}((u_p)_p) := \Phi((v_p)_p, \tfrac{1}{u}). One can show \tilde{\Phi} is well-defined, \Z-linear, satisfies \Phi = \tilde{\Phi} \circ \Psi and is unique with these properties. :Corollary. Define \mathbb{A}_\Z := \widehat{\Z} \times \R. This results in an algebraic isomorphism \mathbb{A}_{\Q} \cong \mathbb{A}_{\Z}\otimes_{\Z} \Q. Proof. \mathbb{A}_\Z \otimes_\Z \Q = \left (\widehat{\Z}\times \R \right )\otimes_\Z \Q \cong \left (\widehat{\Z} \otimes_\Z \Q \right )\times (\R \otimes_\Z \Q) \cong \left (\widehat{\Z}\otimes_{\Z} \Q \right ) \times \R = \mathbb{A}_{\Q,\text{fin}} \times \R = \mathbb{A}_{\Q}. :Lemma. For a number field K, \mathbb{A}_K=\mathbb{A}_{\Q}\otimes_{\Q} K. Remark. Using \mathbb{A}_{\Q}\otimes_{\Q} K \cong \mathbb{A}_{\Q} \oplus \dots \oplus \mathbb{A}_{\Q}, where there are [K:\Q] summands, give the right side receives the product topology and transport this topology via the isomorphism onto \mathbb{A}_{\Q}\otimes_{\Q} K. The adele ring of a finite extension If L/K be a finite extension, then L is a global field. Thus \mathbb{A}_L is defined, and \textstyle \mathbb{A}_L= {\prod_v}^' L_v. \mathbb{A}_K can be identified with a subgroup of \mathbb{A}_L. Map a=(a_v)_v \in \mathbb{A}_K to a'=(a'_w)_w \in \mathbb{A}_L where a'_w=a_v \in K_v \subset L_w for w|v. Then a=(a_w)_w \in \mathbb{A}_L is in the subgroup \mathbb{A}_K, 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. :Lemma. If L/K is a finite extension, then \mathbb{A}_L\cong\mathbb{A}_K \otimes_K L both algebraically and topologically. With the help of this isomorphism, the inclusion \mathbb{A}_K \subset \mathbb{A}_L is given by :\begin{cases} \mathbb{A}_K \to \mathbb{A}_L\\ \alpha \mapsto \alpha \otimes_K 1 \end{cases} Furthermore, the principal adeles in \mathbb{A}_K can be identified with a subgroup of principal adeles in \mathbb{A}_L via the map :\begin{cases} K \to (K \otimes_K L) \cong L\\ \alpha \mapsto 1 \otimes_K \alpha \end{cases} Proof. Let \omega_1,\ldots, \omega_n be a basis of L over K. Then for almost all v, :\widetilde{O_v} \cong O_v\omega_1 \oplus \cdots \oplus O_v \omega_n. Furthermore, there are the following isomorphisms: : K_v\omega_1 \oplus \cdots \oplus K_v \omega_n \cong K_v \otimes_K L \cong L_v=\prod\nolimits_{w | v} L_w For the second use the map: :\begin{cases} K_v \otimes_K L \to L_v \\\alpha_v \otimes a \mapsto (\alpha_v \cdot (\tau_w(a)))_w \end{cases} in which \tau_w : L \to L_w is the canonical embedding and w | v. The restricted product is taken on both sides with respect to \widetilde{O_v}: : \begin{align} \mathbb{A}_K \otimes_K L &= \left ( {\prod_v}^' K_v \right ) \otimes_K L\\ &\cong {\prod_v}^' (K_v\omega_1 \oplus \cdots \oplus K_v \omega_n)\\ &\cong {\prod_v}^' (K_v \otimes_K L)\\ &\cong {\prod_v}^' L_v \\ &=\mathbb{A}_L \end{align} :Corollary. As additive groups \mathbb{A}_L \cong \mathbb{A}_K \oplus \cdots \oplus \mathbb{A}_K, where the right side has [L:K] summands. The set of principal adeles in \mathbb{A}_L is identified with the set K \oplus \cdots \oplus K, where the left side has [L:K] summands and K is considered as a subset of \mathbb{A}_K. The adele ring of vector-spaces and algebras :Lemma. Suppose P\supset P_{\infty} is a finite set of places of K and define ::\mathbb{A}_K(P):=\prod_{v \in P} K_v \times \prod_{v \notin P} O_v. :Equip \mathbb{A}_K(P) with the product topology and define addition and multiplication component-wise. Then \mathbb{A}_K(P) is a locally compact topological ring. Remark. If P' is another finite set of places of K containing P then \mathbb{A}_K(P) is an open subring of \mathbb{A}_K(P'). Now, an alternative characterisation of the adele ring can be presented. The adele ring is the union of all sets \mathbb{A}_K(P): :\mathbb{A}_K = \bigcup_{P \supset P_\infty, |P| Equivalently \mathbb{A}_K is the set of all x=(x_v)_v so that |x_v|_v \leq 1 for almost all v The topology of \mathbb{A}_K is induced by the requirement that all \mathbb{A}_K(P) be open subrings of \mathbb{A}_K. Thus, \mathbb{A}_K 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 :\mathbb{A}_K'(P,v) := \prod_{w \in P \setminus \{v\}} K_w \times \prod_{w \notin P} O_w. Then: :\mathbb{A}_K(P) \cong K_v \times \mathbb{A}_K'(P,v). Furthermore, define :\mathbb{A}_K'(v):=\bigcup_{P \supset P_{\infty} \cup \{v\}} \mathbb{A}_K'(P,v), where P runs through all finite sets containing P_{\infty} \cup \{v\}. Then: :\mathbb{A}_K \cong K_v \times \mathbb{A}_K'(v), via the map (a_w)_w \mapsto (a_v, (a_w)_{w \neq v}). The entire procedure above holds with a finite subset \widetilde{P} instead of \{v\}. By construction of \mathbb{A}_K'(v), there is a natural embedding: K_v \hookrightarrow \mathbb{A}_K. Furthermore, there exists a natural projection \mathbb{A}_K \twoheadrightarrow K_v. The adele ring of a vector-space Let E be a finite dimensional vector-space over K and \{\omega_1,\ldots,\omega_n\} a basis for E over K. For each place v of K: :\begin{align} E_v &:=E \otimes_K K_v \cong K_v\omega_1 \oplus \cdots \oplus K_v\omega_n \\ \widetilde{O_v} &:=O_v\omega_1 \oplus \cdots \oplus O_v\omega_n \end{align} The adele ring of E is defined as :\mathbb{A}_E:= {\prod_v}^' E_v. 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, \mathbb{A}_E is equipped with the restricted product topology. Then \mathbb{A}_E = E \otimes_K \mathbb{A}_K and E is embedded in \mathbb{A}_E naturally via the map e \mapsto e \otimes 1. An alternative definition of the topology on \mathbb{A}_E can be provided. Consider all linear maps: E \to K. Using the natural embeddings E \to \mathbb{A}_E and K \to \mathbb{A}_K, extend these linear maps to: \mathbb{A}_E \to \mathbb{A}_K. The topology on \mathbb{A}_E 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 E \cong K^n. Therefore fixing a basis induces an isomorphism (\mathbb{A}_K)^n \cong \mathbb{A}_E. 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 :\begin{align} \mathbb{A}_E &= E \otimes_K \mathbb{A}_K\\ &\cong (K \otimes_K \mathbb{A}_K) \oplus \cdots \oplus (K \otimes_K \mathbb{A}_K)\\ &\cong \mathbb{A}_K \oplus \cdots \oplus \mathbb{A}_K \end{align} 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, \mathbb{A}_{A} is defined and \mathbb{A}_A \cong \mathbb{A}_K \otimes_K A. Since there is multiplication on \mathbb{A}_K and A, a multiplication on \mathbb{A}_A can be defined via: :\forall \alpha, \beta \in \mathbb{A}_K \text{ and } \forall a,b \in A: \qquad (\alpha \otimes_K a) \cdot (\beta \otimes_K b):=(\alpha\beta)\otimes_K(ab). As a consequence, \mathbb{A}_{A} is an algebra with a unit over \mathbb{A}_K. 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, P\supset P_{\infty}, define :\mathbb{A}_{A}(P,\alpha) =\prod_{v \in P} A_v \times \prod_{v \notin P} M_v. One can show there is a finite set P_0, so that \mathbb{A}_{A}(P,\alpha) is an open subring of \mathbb{A}_{A}, if P \supset P_0. Furthermore \mathbb{A}_{A} 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 \mathbb{A}_K=\mathbb{A}_K \otimes_K K and \mathbb{A}_L=\mathbb{A}_K \otimes_K L from the Lemma above, \mathbb{A}_K can be interpreted as a closed subring of \mathbb{A}_L. For this embedding, write \operatorname{con}_{L/K}. Explicitly for all places w of L above v and for any \alpha \in \mathbb{A}_K, (\operatorname{con}_{L/K}(\alpha))_w=\alpha_v \in K_v. Let M/L/K be a tower of global fields. Then: :\operatorname{con}_{M/K}(\alpha)=\operatorname{con}_{M/L}(\operatorname{con}_{L/K}(\alpha)) \qquad \forall \alpha \in \mathbb{A}_K. Furthermore, restricted to the principal adeles \operatorname{con} is the natural injection K \to L. Let \{\omega_1,\ldots,\omega_n\} be a basis of the field extension L/K. Then each \alpha \in \mathbb{A}_L can be written as \textstyle \sum_{j=1}^n \alpha_j \omega_j, where \alpha_j \in \mathbb{A}_K are unique. The map \alpha \mapsto \alpha_j is continuous. Define \alpha_{ij} depending on \alpha via the equations: :\begin{align} \alpha \omega_1 &=\sum_{j=1}^n \alpha_{1j} \omega_j \\ &\vdots \\ \alpha \omega_n &=\sum_{j=1}^n \alpha_{nj} \omega_j \end{align} Now, define the trace and norm of \alpha as: :\begin{align} \operatorname{Tr}_{L/K}(\alpha) &:= \operatorname{Tr} ((\alpha_{ij})_{i,j})=\sum_{i=1}^n \alpha_{ii}\\ N_{L/K}(\alpha) &:= N ((\alpha_{ij})_{i,j})=\det((\alpha_{ij})_{i,j}) \end{align} These are the trace and the determinant of the linear map :\begin{cases} \mathbb{A}_L \to \mathbb{A}_L \\ x \mapsto \alpha x\end{cases} They are continuous maps on the adele ring, and they fulfil the usual equations: : \begin{align} \operatorname{Tr}_{L/K}(\alpha+\beta)&=\operatorname{Tr}_{L/K}(\alpha) + \operatorname{Tr}_{L/K}(\beta) && \forall \alpha, \beta \in \mathbb{A}_L\\ \operatorname{Tr}_{L/K}(\operatorname{con}(\alpha))&=n\alpha && \forall \alpha \in \mathbb{A}_K\\ N_{L/K}(\alpha \beta)&=N_{L/K}(\alpha) N_{L/K}(\beta) && \forall \alpha, \beta \in \mathbb{A}_L\\ N_{L/K}(\operatorname{con}(\alpha))&=\alpha^n && \forall \alpha \in \mathbb{A}_K \end{align} Furthermore, for \alpha \in L, \operatorname{Tr}_{L/K}(\alpha) and N_{L/K}(\alpha) are identical to the trace and norm of the field extension L/K. For a tower of fields M/L/K, the result is: : \begin{align} \operatorname{Tr}_{L/K}(\operatorname{Tr}_{M/L}(\alpha)) &= \operatorname{Tr}_{M/K}(\alpha) && \forall \alpha \in \mathbb{A}_M\\ N_{L/K} (N_{M/L}(\alpha))&=N_{M/K}(\alpha) && \forall \alpha \in \mathbb{A}_M \end{align} Moreover, it can be proven that: :\begin{align} \operatorname{Tr}_{L/K}(\alpha) &= \left (\sum_{w | v}\operatorname{Tr}_{L_w/K_v}(\alpha_w) \right )_v && \forall \alpha \in \mathbb{A}_L\\ N_{L/K}(\alpha) &= \left (\prod_{w | v}N_{L_w/K_v}(\alpha_w) \right )_v && \forall \alpha \in \mathbb{A}_L \end{align} Properties of the adele ring :Theorem. For every set of places S, \mathbb{A}_{K,S} is a locally compact topological ring. Remark. The result above also holds for the adele ring of vector-spaces and algebras over K. :Theorem. K is discrete and cocompact in \mathbb{A}_K. In particular, K is closed in \mathbb{A}_K. Proof. Prove the case K=\Q. To show \Q\subset \mathbb{A}_\Q 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:= \left \{ (\alpha_p)_p \left | \forall p U is an open neighbourhood of 0 \in \mathbb{A}_\Q. It is claimed that U \cap \Q = \{0\}. Let \beta \in U \cap \Q, then \beta \in \Q and |\beta|_p \leq 1 for all p and therefore \beta \in \Z. Additionally, \beta \in (-1,1) and therefore \beta=0. Next, to show compactness, define: :W:= \left \{(\alpha_p)_p \left | \forall p Each element in \mathbb{A}_\Q /\Q has a representative in W, that is for each \alpha \in \mathbb{A}_\Q, there exists \beta \in \Q such that \alpha - \beta \in W. Let \alpha=(\alpha_p)_p \in \mathbb{A}_\Q, be arbitrary and p be a prime for which |\alpha_p|>1. Then there exists r_p=z_p/p^{x_p} with z_p \in \Z, x_p \in \N and |\alpha_p-r_p|\leq 1. Replace \alpha with \alpha-r_p and let q \neq p be another prime. Then: :\left |\alpha_q-r_p \right |_q \leq \max \left \{|a_q|_q,|r_p|_q \right \} \leq \max \left \{|a_q|_q,1 \right \} \leq 1. Next, it can be claimed that: :|\alpha_q-r_p|_q \leq 1 \Longleftrightarrow |\alpha_q|_q \leq 1. 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 \alpha-r \in \widehat{\Z} \times \R. Now select s \in \Z such that \alpha_\infty-r-s \in \left [-\tfrac{1}{2}, \tfrac{1}{2} \right ]. Then \alpha-(r+s) \in W. The continuous projection \pi:W \to\mathbb{A}_\Q /\Q is surjective, therefore \mathbb{A}_\Q /\Q, as the continuous image of a compact set, is compact. :Corollary. Let E be a finite-dimensional vector-space over K. Then E is discrete and cocompact in \mathbb{A}_E. :Theorem. The following are assumed: :*\mathbb{A}_{\Q}= \Q +\mathbb{A}_{\Z}. :*\Z =\Q \cap \mathbb{A}_{\Z}. :*\mathbb{A}_{\Q}/\Z is a divisible group. :*\Q \subset \mathbb{A}_{\Q,\text{fin}} is dense. Proof. The first two equations can be proved in an elementary way. By definition \mathbb{A}_{\Q}/\Z is divisible if for any n \in \N and y \in \mathbb{A}_{\Q}/\Z the equation nx=y has a solution x \in \mathbb{A}_{\Q}/\Z. It is sufficient to show \mathbb{A}_{\Q} is divisible but this is true since \mathbb{A}_{\Q} is a field with positive characteristic in each coordinate. For the last statement note that \mathbb{A}_{\Q,\text{fin}}=\Q \widehat{\Z}, because the finite number of denominators in the coordinates of the elements of \mathbb{A}_{\Q,\text{fin}} can be reached through an element q \in \Q. As a consequence, it is sufficient to show \Z \subset \widehat{\Z} is dense, that is each open subset V \subset \widehat{\Z} contains an element of \Z. Without loss of generality, it can be assumed that :V=\prod_{p \in E} \left(a_p+p^{l_p}\Z_p \right ) \times \prod_{p \notin E}\Z_p, because (p^m\Z_p)_{m \in \N} is a neighbourhood system of 0 in \Z_p. By Chinese Remainder Theorem there exists l \in \Z such that l \equiv a_p \bmod p^{l_p}. Since powers of distinct primes are coprime, l \in V follows. Remark. \mathbb{A}_{\Q}/\Z is not uniquely divisible. Let y=(0,0,\ldots)+\Z \in \mathbb{A}_{\Q}/\Z and n \geq 2 be given. Then :\begin{align} x_1 &=(0,0,\ldots)+\Z \\ x_2 &= \left (\tfrac{1}{n}, \tfrac{1}{n}, \ldots \right )+\Z \end{align} both satisfy the equation nx=y and clearly x_1 \neq x_2 (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 \mathbb{A}_{\Q}/\Z 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 f: \mathbb{A}_K \to \C is called simple if \textstyle f=\prod_v f_v, where f_v:K_v \to \C are measurable and f_v= \mathbf{1}_{O_v} for almost all v. :Theorem. Since \mathbb{A}_K is a locally compact group with addition, there is an additive Haar measure dx on \mathbb{A}_K. This measure can be normalised such that every integrable simple function \textstyle f=\prod_v f_v satisfies: ::\int_{\mathbb{A}_K} f \, dx = \prod_v \int_{K_v} f_v \, dx_v, :where for v is the measure on K_v such that O_v has unit measure and dx_{\infty} is the Lebesgue measure. The product is finite, i.e., almost all factors are equal to one. ==The idele group==

Definition. Define the idele group of K as the group of units of the adele ring of K, that is I_K := \mathbb{A}_K^{\times}. 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 \mathbb{A}_K 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 \mathbb{A}_{\Q} is not continuous. The sequence :\begin{align} x_1&=(2,1,\ldots)\\ x_2&=(1,3,1,\ldots)\\ x_3&=(1,1,5,1,\ldots)\\ &\vdots \end{align} converges to 1 \in \mathbb{A}_{\Q}. To see this let U be neighbourhood of 0, without loss of generality it can be assumed: :U=\prod_{p \leq N} U_p \times \prod_{p > N}\Z_p Since (x_n)_p-1 \in \Z_p 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 \mathbb{A}_{\Q}. :Lemma. Let R be a topological ring. Define: ::\begin{cases} \iota: R^{\times} \to R \times R\\ x \mapsto (x,x^{-1}) \end{cases} :Equipped with the topology induced from the product on topology on R \times R and \iota, R^{\times} is a topological group and the inclusion map R^{\times} \subset R is continuous. It is the coarsest topology, emerging from the topology on R, that makes R^\times a topological group. Proof. Since R is a topological ring, it is sufficient to show that the inverse map is continuous. Let U\subset R^\times be open, then U \times U^{-1} \subset R \times R is open. It is necessary to show U^{-1} \subset R^\times is open or equivalently, that U^{-1}\times (U^{-1})^{-1}=U^{-1} \times U \subset R \times R 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: I_{K,S}:=\mathbb{A}_{K,S}^\times, I_K^S:=(\mathbb{A}_{K}^S)^{\times}. :Lemma. The following identities of topological groups hold: ::\begin{align} I_{K,S}&= {\prod_{v \in S}}^' K_v^{\times}\\ I_K^S&= {\prod_{v \notin S}}^' K_v^{\times}\\ I_K&= {\prod_v}^' K_v^{\times} \end{align} :where the restricted product has the restricted product topology, which is generated by restricted open rectangles of the form ::\prod_{v \in E} U_v \times \prod_{v \notin E} O_v^{\times}, :where E is a finite subset of the set of all places and U_v \subset K_v^{\times} are open sets. Proof. Prove the identity for I_K; the other two follow similarly. First show the two sets are equal: :\begin{align} I_K &=\{x=(x_v)_v \in \mathbb{A}_K: \exists y=(y_v)_v \in \mathbb{A}_K: xy=1\}\\ &=\{x=(x_v)_v \in \mathbb{A}_K: \exists y=(y_v)_v \in \mathbb{A}_K: x_v \cdot y_v =1 \quad \forall v\} \\ &=\{x=(x_v)_v: x_v \in K_v^\times \forall v \text{ and } x_v \in O_v^\times \text{ for almost all } v\} \\ &= {\prod_v}^' K_v^\times \end{align} In going from line 2 to 3, x as well as x^{-1}=y have to be in \mathbb{A}_K, meaning x_v \in O_v for almost all v and x_v^{-1} \in O_v for almost all v. Therefore, x_v \in O_v^\times 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 U \subset I_K, which is open in the topology of the idele group, meaning U \times U^{-1} \subset \mathbb{A}_K \times \mathbb{A}_K 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. :Lemma. For each set of places, S, I_{K,S} is a locally compact topological group. 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 1 \in \mathbb{A}_K(P_\infty)^{\times} is a neighbourhood system of 1 \in I_K. Alternatively, take all sets of the form: :\prod_v U_v, where U_v is a neighbourhood of 1 \in K_v^\times and U_v=O_v^\times for almost all v. Since the idele group is locally compact, there exists a Haar measure d^\times x on it. This can be normalised, so that :\int_{I_{K,\text{fin}}} \mathbf{1}_{\widehat{O}}\, d^\times x =1. This is the normalisation used for the finite places. In this equation, I_{K,\text{fin}} is the finite idele group, meaning the group of units of the finite adele ring. For the infinite places, use the multiplicative Lebesgue measure \tfrac{dx}. The idele group of a finite extension :Lemma. Let L/K be a finite extension. Then: ::I_L= {\prod_v}^' L_v^\times. :where the restricted product is with respect to \widetilde{O_v}^{\times}. :Lemma. There is a canonical embedding of I_K in I_L. Proof. Map a=(a_v)_v \in I_K to a'=(a'_w)_w \in I_L with the property a'_w=a_v \in K_v^\times \subset L_w^\times for w|v. Therefore, I_K can be seen as a subgroup of I_L. An element a=(a_w)_w \in I_L is in this subgroup if and only if his components satisfy the following properties: a_w \in K_v^\times 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 \mathbb{A}_A^{\times} is not a topological group with the subset-topology in general, equip \mathbb{A}_{A}^{\times} with the topology similar to I_K above and call \mathbb{A}_A^{\times} the idele group. The elements of the idele group are called idele of A. :Proposition. Let \alpha be a finite subset of A, containing a basis of A over K. For each finite place v of K, let \alpha_v be the O_v-module generated by \alpha in A_v. There exists a finite set of places P_0 containing P_{\infty} such that for all v \notin P_0,\alpha_v is a compact subring of A_v. Furthermore, \alpha_v contains A_v^\times. For each v, A_v^{\times} is an open subset of A_v and the map x \mapsto x^{-1} is continuous on A_v^{\times}. As a consequence x \mapsto (x,x^{-1}) maps A_v^{\times} homeomorphically on its image in A_v \times A_v. For each v \notin P_0, the \alpha_v^{\times} are the elements of A_v^\times, mapping in \alpha_v \times \alpha_v with the function above. Therefore, \alpha_v^{\times} is an open and compact subgroup of A_v^\times. Alternative characterisation of the idele group :Proposition. Let P \supset P_{\infty} be a finite set of places. Then ::\mathbb{A}_{A}(P,\alpha)^{\times}:=\prod_{v \in P} A_v^{\times} \times \prod_{v \notin P} \alpha_v^{\times} :is an open subgroup of \mathbb{A}_{A}^{\times}, where \mathbb{A}_{A}^{\times} is the union of all \mathbb{A}_{A}(P,\alpha)^{\times}. :Corollary. In the special case of A=K for each finite set of places P \supset P_{\infty}, ::\mathbb{A}_K(P)^{\times}=\prod_{v \in P}K_v^{\times} \times \prod_{v \notin P}O_v^{\times} :is an open subgroup of \mathbb{A}_K^{\times}=I_K. Furthermore, I_K is the union of all \mathbb{A}_K(P)^{\times}. Norm on the idele group It is possible to transfer the norm from the adele ring to the idele group. Let \alpha \in I_K. Then \operatorname{con}_{L/K}(\alpha) \in I_L and therefore there is an injective group homomorphism :\operatorname{con}_{L/K}: I_K \hookrightarrow I_L. Since \alpha \in I_L, it is invertible, N_{L/K}(\alpha) is invertible too, because (N_{L/K}(\alpha))^{-1}= N_{L/K}(\alpha^{-1}). Therefore N_{L/K}(\alpha) \in I_K. As a consequence, the restriction of the norm-function induces a continuous function: :N_{L/K}: I_L \to I_K. The idele class group :Lemma. There is natural embedding of K^{\times} into I_{K,S} given by the diagonal map: a \mapsto (a,a,a,\ldots). Proof. Since K^{\times} is a subset of K_v^{\times} for all v, the embedding is well-defined and injective. :Corollary. A^{\times} is a discrete subgroup of \mathbb{A}_{A}^{\times}. Definition. 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 C_K := I_K/K^{\times} 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. :Lemma. Let L/K be a finite extension. The embedding I_K \to I_L induces an injective map: ::\begin{cases} C_K \to C_L\\ \alpha K^\times \mapsto \alpha L^\times \end{cases} Properties of the idele group Absolute value on the idele group of K and 1-idele Definition. For \alpha=(\alpha_v)_v \in I_K define: \textstyle |\alpha|:=\prod_v |\alpha_v|_v. Since \alpha is an idele this product is finite and therefore well-defined. Remark. The definition can be extended to \mathbb{A}_K by allowing infinite products. However, these infinite products vanish and so |\cdot| vanishes on \mathbb{A}_K \setminus I_K. |\cdot| will be used to denote both the function on I_K and \mathbb{A}_K. :Theorem. |\cdot|:I_K \to \R_+ is a continuous group homomorphism. Proof. Let \alpha, \beta \in I_K. : \begin{align} &=\prod_v|\alpha_v \cdot \beta_v|_v\\ &=\prod_v(|\alpha_v|_v \cdot |\beta_v|_v)\\ &=\left(\prod_v |\alpha_v|_v\right) \cdot \left(\prod_v |\beta_v|_v\right)\\ &= |\alpha|\cdot |\beta| \end{align} 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:=\{x \in I_K: |x|=1\}=\ker(|\cdot|). I_K^1 is a subgroup of I_K. Since I_K^1=|\cdot|^{-1}(\{1\}), it is a closed subset of \mathbb{A}_K. Finally the \mathbb{A}_K-topology on I_K^1 equals the subset-topology of I_K on I_K^1. :'''Artin's Product Formula.''' |k|=1 for all k \in K^\times. 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 a \in K^\times. It has to be shown that: :\prod_v|a|_v=1. For finite place v for which the corresponding prime ideal \mathfrak{p}_v does not divide (a), v(a)=0 and therefore |a|_v=1. This is valid for almost all \mathfrak{p}_v. There is: : \begin{align} \prod_v|a|_v&=\prod_{p \leq \infty} \prod_{v| p}|a|_v\\ &=\prod_{p \leq \infty} \prod_{v| p}|N_{K_v/ \Q_p}(a)|_p\\ &=\prod_{p \leq \infty} |N_{K /\Q}(a)|_p \end{align} In going from line 1 to line 2, the identity |a|_w=|N_{L_w/ K_v}(a)|_v, 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: :a=\pm\prod_{p 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 |\cdot|_{\infty} or a p-adic absolute value. Therefore: :\begin{align} &= \left(\prod_{p :Lemma. There exists a constant C, depending only on K, such that for every \alpha=(\alpha_v)_v \in \mathbb{A}_K satisfying \textstyle \prod_v |\alpha_v|_v > C, there exists \beta \in K^{\times} such that |\beta_v|_v\leq |\alpha_v|_v for all v. :Corollary. Let v_0 be a place of K and let \delta_v > 0 be given for all v \neq v_0 with the property \delta_v=1 for almost all v. Then there exists \beta \in K^{\times}, so that |\beta| \leq \delta_v for all v \neq v_0. Proof. Let C be the constant from the lemma. Let \pi_v be a uniformising element of O_v. Define the adele \alpha=(\alpha_v)_v via \alpha_v:=\pi_v^{k_v} with k_v \in \Z minimal, so that |\alpha_v|_v \leq \delta_v for all v \neq v_0. Then k_v=0 for almost all v. Define \alpha_{v_0}:=\pi_{v_0}^{k_{v_0}} with k_{v_0}\in \Z, so that \textstyle \prod_v |\alpha_v|_v > C. This works, because k_v=0 for almost all v. By the Lemma there exists \beta \in K^\times, so that |\beta|_v \leq |\alpha_v|_v \leq \delta_v for all v \neq v_0. :Theorem. K^\times is discrete and cocompact in I_K^1. Proof. Since K^\times is discrete in I_K it is also discrete in I_K^1. To prove the compactness of I_K^1/K^\times let C is the constant of the Lemma and suppose \alpha \in \mathbb{A}_K satisfying \textstyle \prod_v |\alpha_v|_v > C is given. Define: :W_\alpha:= \left \{\xi=(\xi_v)_v \in \mathbb{A}_K | |\xi_v|_v\leq |\alpha_v|_v \text{ for all } v\right \}. Clearly W_\alpha is compact. It can be claimed that the natural projection W_{\alpha} \cap I_K^1 \to I_K^1/K^\times is surjective. Let \beta=(\beta_v)_v \in I_K^1 be arbitrary, then: :|\beta| = \prod_v |\beta_v|_v=1, and therefore :\prod_v |\beta_v^{-1}|_v=1. It follows that :\prod_v |\beta_v^{-1}\alpha_v|_v=\prod_v |\alpha_v|_v>C. By the Lemma there exists \eta \in K^\times such that |\eta|_v \leq |\beta_v^{-1}\alpha_v|_v for all v, and therefore \eta\beta \in W_{\alpha} proving the surjectivity of the natural projection. Since it is also continuous the compactness follows. :Theorem. There is a canonical isomorphism I_{\Q}^1/\Q^\times \cong \widehat{\Z}^\times. Furthermore, \widehat{\Z}^\times \times \{1\} \subset I_{\Q}^1 is a set of representatives for I_{\Q}^1/\Q^\times and \widehat{\Z}^\times \times (0, \infty) \subset I_{\Q} is a set of representatives for I_{\Q}/\Q^\times. Proof. Consider the map :\begin{cases} \phi: \widehat{\Z}^\times \to I_{\Q}^1/\Q^\times \\ (a_p)_p \mapsto ((a_p)_p,1)\Q^\times \end{cases} This map is well-defined, since |a_p|_p=1 for all p and therefore \textstyle \left(\prod_{p Obviously \phi is a continuous group homomorphism. Now suppose ((a_p)_p,1)\Q^\times=((b_p)_p,1)\Q^\times. Then there exists q \in \Q^\times such that ((a_p)_p,1)q=((b_p)_p,1). By considering the infinite place it can be seen that q=1 proves injectivity. To show surjectivity let ((\beta_p)_p, \beta_\infty) \Q^\times\in I_{\Q}^1/\Q^\times. The absolute value of this element is 1 and therefore :|\beta_\infty|_\infty=\frac{1}{\prod_p |\beta_p|_p} \in \Q. Hence \beta_\infty \in \Q and there is: :((\beta_p)_p, \beta_\infty) \Q^\times= \left ( \left (\frac{\beta_p}{\beta_\infty} \right )_p,1 \right )\Q^\times. Since :\forall p: \qquad \left | \frac{\beta_p}{\beta_\infty} \right |_p=1, It has been concluded that \phi is surjective. :Theorem. Let K be a number field. There exists a finite set of places S, such that: ::I_K= \left (I_{K,S} \times \prod_{v \notin S} O_v^\times \right ) K^\times= \left(\prod_{v \in S} K_v^\times \times \prod_{v \notin S} O_v^\times\right) K^\times. Proof. The class number of a number field is finite so let \mathfrak{a}_1, \ldots, \mathfrak{a}_h be the ideals, representing the classes in \operatorname{Cl}_K. These ideals are generated by a finite number of prime ideals \mathfrak{p}_1, \ldots, \mathfrak{p}_n. Let S be a finite set of places containing P_\infty and the finite places corresponding to \mathfrak{p}_1, \ldots, \mathfrak{p}_n. Consider the isomorphism: :I_K/ \left(\prod_{v induced by :(\alpha_v)_v \mapsto \prod_{v At infinite places the statement is immediate, so the statement has been proved for finite places. The inclusion ″\supset″ is obvious. Let \alpha \in I_{K,\text{fin}}. The corresponding ideal \textstyle (\alpha)=\prod_{v belongs to a class \mathfrak{a}_iK^{\times}, meaning (\alpha)=\mathfrak{a}_i(a) for a principal ideal (a). The idele \alpha'=\alpha a^{-1} maps to the ideal \mathfrak{a}_i under the map I_{K,\text{fin}} \to J_K. That means \textstyle \mathfrak{a}_i=\prod_{v Since the prime ideals in \mathfrak{a}_i are in S, it follows v(\alpha'_v)=0 for all v \notin S, that means \alpha'_v \in O_v^\times for all v \notin S. It follows, that \alpha'=\alpha a^{-1} \in I_{K,S}, therefore \alpha \in I_{K,S}K^\times. ==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: :Theorem (finiteness of the class number of a number field). Let K be a number field. Then |\operatorname{Cl}_K| Proof. The map :\begin{cases} I_K^1 \to J_K \\ \left ((\alpha_v)_{v is surjective and therefore \operatorname{Cl}_K is the continuous image of the compact set I_K^1/K^{\times}. Thus, \operatorname{Cl}_K 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 P \supset P_{\infty} be a finite set of places. Define :\begin{align} \Omega(P)&:=\prod_{v\in P} K_v^\times \times \prod_{v \notin P} O_v^{\times}=(\mathbb{A}_K(P))^\times\\ E(P)&:=K^{\times} \cap \Omega(P) \end{align} Then E(P) is a subgroup of K^{\times}, containing all elements \xi \in K^{\times} 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 \Omega_1(P):=\Omega(P)\cap I_K^1. An alternative definition is: E(P)=K(P)^{\times}, where K(P) is a subring of K defined by :K(P):= K \cap \left(\prod_{v\in P} K_v \times \prod_{v \notin P}O_v\right). As a consequence, K(P) contains all elements \xi \in K, which fulfil v(\xi) \geq 0 for all v \notin P. :Lemma 1. Let 0 The following set is finite: ::\left \{\eta \in E(P): \left. \begin{cases}|\eta_v|_v = 1 & \forall v \notin P \\ c \leq |\eta_v|_v \leq C & \forall v \in P \end{cases} \right \} \right \}. Proof. Define :W:= \left \{(\alpha_v)_v: \left. \begin{cases}|\alpha_v|_v= 1 & \forall v \notin P \\ c \leq |\alpha_v|_v \leq C & \forall v \in P \end{cases} \right \} \right \}. 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. :Lemma 2. Let E be set of all \xi \in K such that |\xi|_v =1 for all v. Then E = \mu(K), the group of all roots of unity of K. In particular it is finite and cyclic. Proof. All roots of unity of K have absolute value 1 so \mu(K) \subset E. For converse note that Lemma 1 with c=C=1 and any P implies E is finite. Moreover E \subset E(P) for each finite set of places P \supset P_\infty. Finally suppose there exists \xi \in E, which is not a root of unity of K. Then \xi^n \neq 1 for all n \in \N contradicting the finiteness of E. :Unit Theorem. E(P) is the direct product of E and a group isomorphic to \Z^s, where s=0, if P= \emptyset and s=|P|-1, if P \neq \emptyset. :'''Dirichlet's Unit Theorem.''' Let K be a number field. Then O^\times\cong\mu(K) \times \Z^{r+s-1}, where \mu(K) is the finite cyclic group of all roots of unity of K, r is the number of real embeddings of K and s is the number of conjugate pairs of complex embeddings of K. It stands, that [K:\Q]=r+2s. 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 |P_\infty|=r+s. Then there is: : \begin{align} E \times \Z^{r+s-1} = E(P_\infty)&=K^\times \cap \left(\prod_{v | \infty} K_v^\times \times \prod_{v Approximation theorems :Weak Approximation Theorem. Let |\cdot|_1, \ldots, |\cdot|_N, be inequivalent valuations of K. Let K_n be the completion of K with respect to |\cdot|_n. Embed K diagonally in K_1 \times \cdots \times K_N. Then K is everywhere dense in K_1 \times \cdots \times K_N. In other words, for each \varepsilon > 0 and for each (\alpha_1, \ldots, \alpha_N) \in K_1 \times \cdots \times K_N, there exists \xi \in K, such that: ::\forall n \in \{ 1, \ldots, N\}: \quad |\alpha_n - \xi|_n :Strong Approximation Theorem. Let v_0 be a place of K. Define ::V:= {\prod_{v \neq v_0}}^' K_v. :Then K is dense in V. 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 :Hasse-Minkowski Theorem. A quadratic form on K is zero, if and only if, the quadratic form is zero in each completion K_v. 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 \widehat{G}. Equivalently \widehat{G} is the set of all continuous group homomorphisms from G to \mathbb{T}:=\{z \in \C :|z|=1\}. 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. :Theorem. The adele ring is self-dual: \mathbb{A}_K\cong \widehat{\mathbb{A}_K}. 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: :\begin{cases} e_\infty: \R \to \mathbb{T} \\ e_\infty(t) :=\exp(2\pi it) \end{cases} Then the following map is an isomorphism which respects topologies: :\begin{cases} \phi: \R \to \widehat{\R} \\ s \mapsto \begin{cases} \phi_s: \R \to \mathbb{T} \\ \phi_s(t) := e_\infty(ts) \end{cases}\end{cases} :Theorem (algebraic and continuous duals of the adele ring). Let \chi be a non-trivial character of \mathbb{A}_K, which is trivial on K. Let E be a finite-dimensional vector-space over K. Let E^\star and \mathbb{A}_E^\star be the algebraic duals of E and \mathbb{A}_E. Denote the topological dual of \mathbb{A}_E by \mathbb{A}_E' and use \langle \cdot,\cdot \rangle and [{\cdot},{\cdot}] to indicate the natural bilinear pairings on \mathbb{A}_E \times \mathbb{A}_E' and \mathbb{A}_E \times \mathbb{A}_E^{\star}. Then the formula \langle e,e'\rangle = \chi([e,e^\star]) for all e \in \mathbb{A}_E determines an isomorphism e^\star \mapsto e' of \mathbb{A}_E^\star onto \mathbb{A}_E', where e' \in \mathbb{A}_E' and e^\star \in \mathbb{A}_E^\star. Moreover, if e^\star \in \mathbb{A}_E^\star fulfils \chi([e,e^\star])=1 for all e \in E, then e^\star \in E^\star. Tate's thesis With the help of the characters of \mathbb{A}_K, 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, :\int_{\widehat{\Z}} |x|^s d^\times x = \zeta(s), where d^\times x is the unique Haar measure on I_{\Q,\text{fin}} normalised such that \widehat{\Z}^\times 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: :\begin{align} I_{\Q} &= \operatorname{GL} (1, \mathbb{A}_{\Q}) \\ I_{\Q}^1 &= (\operatorname{GL} (1, \mathbb{A}_\Q))^1:=\{x \in \operatorname{GL} (1, \mathbb{A}_\Q): |x|=1\} \\ \Q^{\times} &= \operatorname{GL} (1, \Q) \end{align} Based on these identification a natural generalisation would be to replace the idele group and the 1-idele with: :\begin{align} I_{\Q} &\leftrightsquigarrow \operatorname{GL} (2, \mathbb{A}_{\Q}) \\ I_{\Q}^1 &\leftrightsquigarrow (\operatorname{GL} (2, \mathbb{A}_\Q))^1:=\{x \in \operatorname{GL} (2, \mathbb{A}_\Q): |\det (x) |=1\} \\ \Q &\leftrightsquigarrow \operatorname{GL} (2, \Q) \end{align} And finally :\Q^{\times} \backslash I_{\Q}^1 \cong \Q^{\times} \backslash I_{\Q} \leftrightsquigarrow (\operatorname{GL} (2, \Q) \backslash (\operatorname{GL} (2, \mathbb{A}_\Q))^1 \cong (\operatorname{GL} (2, \Q)Z_{\R}) \backslash\operatorname{GL} (2, \mathbb{A}_\Q), where Z_\R is the centre of \operatorname{GL} (2, \R). Then it define an automorphic form as an element of L^2((\operatorname{GL} (2, \Q)Z_{\R}) \backslash \operatorname{GL} (2, \mathbb{A}_\Q)). In other words an automorphic form is a function on \operatorname{GL} (2, \mathbb{A}_{\Q}) satisfying certain algebraic and analytic conditions. For studying automorphic forms, it is important to know the representations of the group \operatorname{GL} (2, \mathbb{A}_{\Q}). It is also possible to study automorphic L-functions, which can be described as integrals over \operatorname{GL} (2, \mathbb{A}_{\Q}). Generalise even further is possible by replacing \Q with a number field and \operatorname{GL} (2) with an arbitrary reductive algebraic group. Further applications A generalisation of Artin reciprocity law leads to the connection of representations of \operatorname{GL} (2, \mathbb{A}_K) 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. ==References==