Adele ring

In mathematics, the adele ring is a construction in number theory that combines all local versions of a global field into one object. For the rational numbers, these local versions include the real numbers and the fields of ${\displaystyle p}$-adic numbers for all prime numbers ${\displaystyle p}$. More generally, if ${\displaystyle K}$ is a global field, its adele ring, often denoted ${\displaystyle \mathbb {A} _{K}}$, is a topological ring built from the completions ${\displaystyle K_{v}}$ of ${\displaystyle K}$ at all its places ${\displaystyle v}$. Formally, it is a restricted product of the local fields ${\displaystyle K_{v}}$, with respect to the valuation rings at the non-archimedean places. Its elements are called adeles.

The restricted product topology makes ${\displaystyle \mathbb {A} _{K}}$ a locally compact topological ring. The field ${\displaystyle K}$ embeds diagonally in ${\displaystyle \mathbb {A} _{K}}$ as a discrete subring, and the quotient ${\displaystyle \mathbb {A} _{K}/K}$ is compact. As an additive locally compact abelian group, the adele ring is self-dual, making it a natural setting for Fourier analysis on global fields.

The group of units of the adele ring, with its natural topology, is the idele group ${\displaystyle \mathbb {A} _{K}^{\times }}$. The quotient ${\displaystyle \mathbb {A} _{K}^{\times }/K^{\times }}$, called the idele class group, is a central object in class field theory. Adeles and ideles are also used in Tate's thesis, the theory of automorphic forms, local-global principles, and adelic descriptions of divisors, line bundles, and principal bundles on algebraic curves.

Let ${\displaystyle K}$ be a global field, meaning either a number field or a global function field. Let ${\displaystyle v}$ run over the places of ${\displaystyle K}$. For each place ${\displaystyle v}$, let ${\displaystyle K_{v}}$ be the completion of ${\displaystyle K}$ at ${\displaystyle v}$. If ${\displaystyle v}$ is non-archimedean, let ${\displaystyle {\mathcal {O}}_{v}}$ be the corresponding valuation ring.¹²

The set of finite adeles of ${\displaystyle K}$, denoted ${\displaystyle \mathbb {A} _{K,\mathrm {fin} }}$, is the restricted product of the non-archimedean completions ${\displaystyle K_{v}}$ with respect to the subrings ${\displaystyle {\mathcal {O}}_{v}}$:

${\displaystyle \mathbb {A} _{K,\mathrm {fin} }={\prod _{v\nmid \infty }}'K_{v}=\left\{(x_{v})_{v}\in \prod _{v\nmid \infty }K_{v}:x_{v}\in {\mathcal {O}}_{v}{\text{ for all but finitely many }}v\right\}.}$

It is equipped with the restricted product topology. A basis of open sets is given by products

${\displaystyle \prod _{v\in E}U_{v}\times \prod _{v\notin E}{\mathcal {O}}_{v},}$

where ${\displaystyle E}$ is a finite set of non-archimedean places and each ${\displaystyle U_{v}}$ is open in ${\displaystyle K_{v}}$. With componentwise addition and multiplication, ${\displaystyle \mathbb {A} _{K,\mathrm {fin} }}$ is a topological ring.¹³

The adele ring of ${\displaystyle K}$, denoted ${\displaystyle \mathbb {A} _{K}}$, is obtained by adjoining the completions at the archimedean places:

${\displaystyle \mathbb {A} _{K}=\mathbb {A} _{K,\mathrm {fin} }\times \prod _{v\mid \infty }K_{v}={\prod _{v\nmid \infty }}'K_{v}\times \prod _{v\mid \infty }K_{v}.}$

The number of archimedean places is finite, and each archimedean completion is isomorphic to ${\displaystyle \mathbb {R} }$ or ${\displaystyle \mathbb {C} }$. The elements of ${\displaystyle \mathbb {A} _{K}}$ are called adeles of ${\displaystyle K}$. Addition and multiplication are defined componentwise. For brevity, one often writes

${\displaystyle \mathbb {A} _{K}=\prod _{v}'K_{v},}$

with the understanding that the restricted product condition applies only at the non-archimedean places.²¹

If ${\displaystyle K}$ is a global function field, then there are no archimedean places, so ${\displaystyle \mathbb {A} _{K}=\mathbb {A} _{K,\mathrm {fin} }}$.

There is a natural diagonal embedding

${\displaystyle K\hookrightarrow \mathbb {A} _{K},\qquad a\mapsto (a,a,\ldots ).}$

This map is well-defined because an element ${\displaystyle a\in K}$ lies in ${\displaystyle {\mathcal {O}}_{v}}$ for all but finitely many non-archimedean places ${\displaystyle v}$. After this embedding, ${\displaystyle K}$ is regarded as a subring of ${\displaystyle \mathbb {A} _{K}}$, and its elements are sometimes called the principal adeles of ${\displaystyle \mathbb {A} _{K}}$.³¹

More generally, if ${\displaystyle S}$ is a set of places of ${\displaystyle K}$, one may define the ring of ${\displaystyle S}$-adeles by

${\displaystyle \mathbb {A} _{K,S}:=\prod _{v\in S}'K_{v},}$

again using the valuation rings ${\displaystyle {\mathcal {O}}_{v}}$ at the non-archimedean places in ${\displaystyle S}$. If

${\displaystyle \mathbb {A} _{K}^{S}:=\prod _{v\notin S}'K_{v},}$

then there is a natural product decomposition

${\displaystyle \mathbb {A} _{K}\cong \mathbb {A} _{K,S}\times \mathbb {A} _{K}^{S}.}$

The purpose of the adele ring is to look at all completions of a global field ${\displaystyle K}$ at once. For the rational numbers, the usual absolute value gives the completion ${\displaystyle \mathbb {R} }$, but Ostrowski's theorem shows that there are also the ${\displaystyle p}$-adic absolute values, one for each prime number ${\displaystyle p}$. More generally, a global field has a family of completions ${\displaystyle K_{v}}$, one for each place ${\displaystyle v}$. The adele ring packages these completions into a single object, so that analytic methods can be applied while still retaining arithmetic information from all finite primes.¹²

A precursor to this point of view is Minkowski's geometry of numbers. If ${\displaystyle K}$ is a number field with ring of integers ${\displaystyle {\mathcal {O}}_{K}}$, the classical Minkowski embedding places ${\displaystyle {\mathcal {O}}_{K}}$ as a lattice in the finite-dimensional real vector space

${\displaystyle K\otimes _{\mathbb {Q} }\mathbb {R} \cong \mathbb {R} ^{r}\times \mathbb {C} ^{s}.}$

This makes it possible to study arithmetic questions using volume and compactness arguments. The adele ring may be viewed as a local-global enlargement of this construction: instead of using only the archimedean completions, it includes all completions of ${\displaystyle K}$. In the adelic setting, the global field ${\displaystyle K}$ itself embeds diagonally as a discrete subgroup of ${\displaystyle \mathbb {A} _{K}}$, and the quotient ${\displaystyle \mathbb {A} _{K}/K}$ is compact.³¹

The adele ring is defined as a restricted product rather than as the full Cartesian product of all completions. The restricted product condition says that an adele is integral at almost all non-archimedean places. This condition is natural from the point of view of the global field itself: if ${\displaystyle a\in K}$, then ${\displaystyle a}$ belongs to ${\displaystyle {\mathcal {O}}_{v}}$ for all but finitely many finite places ${\displaystyle v}$. Thus the diagonal embedding

${\displaystyle K\hookrightarrow \mathbb {A} _{K}}$

lands in the restricted product.

The restricted product is also the topological condition that makes the adele ring useful for analysis. With its restricted product topology, ${\displaystyle \mathbb {A} _{K}}$ is a locally compact topological ring. Local compactness gives the additive group of ${\displaystyle \mathbb {A} _{K}}$ a Haar measure, making it possible to do harmonic analysis on the adele ring. This is one of the main reasons adeles are useful in modern number theory.¹⁴

Tate's thesis, for example, constructs Fourier analysis on the adele ring and integration over the idele group to give a uniform treatment of Hecke ${\displaystyle L}$-functions. In this approach, global zeta integrals factor into local integrals over the completions ${\displaystyle K_{v}}$, and the local-global structure of the adele ring explains the Euler product, analytic continuation, and functional equation of these ${\displaystyle L}$-functions.⁵⁶

For ${\displaystyle K=\mathbb {Q} }$, Ostrowski's theorem says that the places of ${\displaystyle \mathbb {Q} }$ are given by the usual absolute value and the ${\displaystyle p}$-adic absolute values, one for each prime number ${\displaystyle p}$. The completion at the infinite place is

${\displaystyle \mathbb {Q} _{\infty }=\mathbb {R} ,}$

and the completion at the place corresponding to ${\displaystyle p}$ is the field of ${\displaystyle p}$-adic numbers ${\displaystyle \mathbb {Q} _{p}}$, with valuation ring ${\displaystyle \mathbb {Z} _{p}}$. Thus the adele ring of ${\displaystyle \mathbb {Q} }$ is

${\displaystyle \mathbb {A} _{\mathbb {Q} }=\mathbb {R} \times \prod _{p}'\mathbb {Q} _{p},}$

where the restricted product is taken with respect to the subrings ${\displaystyle \mathbb {Z} _{p}}$. Equivalently,

${\displaystyle \mathbb {A} _{\mathbb {Q} }=\left\{(x_{\infty },x_{2},x_{3},x_{5},\ldots ):x_{\infty }\in \mathbb {R} ,\ x_{p}\in \mathbb {Q} _{p},\ x_{p}\in \mathbb {Z} _{p}{\text{ for all but finitely many }}p\right\}.}$

Thus an adele of ${\displaystyle \mathbb {Q} }$ is a real number together with a ${\displaystyle p}$-adic rational number for each prime ${\displaystyle p}$, such that all but finitely many of the ${\displaystyle p}$-adic components are ${\displaystyle p}$-adic integers.¹²

The finite adeles of ${\displaystyle \mathbb {Q} }$ are

${\displaystyle \mathbb {A} _{\mathbb {Q} ,\mathrm {fin} }=\prod _{p}'\mathbb {Q} _{p}.}$

The integral finite adeles are

${\displaystyle {\widehat {\mathbb {Z} }}=\prod _{p}\mathbb {Z} _{p},}$

the ring of profinite integers. With this notation,

${\displaystyle \mathbb {A} _{\mathbb {Q} }=\mathbb {R} \times \mathbb {A} _{\mathbb {Q} ,\mathrm {fin} },\qquad {\widehat {\mathbb {Z} }}\subset \mathbb {A} _{\mathbb {Q} ,\mathrm {fin} }.}$

The diagonal embedding of ${\displaystyle \mathbb {Q} }$ sends a rational number ${\displaystyle a}$ to the adele

${\displaystyle (a,a,a,\ldots ).}$

This is well-defined because a rational number has only finitely many prime factors in its denominator, so ${\displaystyle a\in \mathbb {Z} _{p}}$ for all but finitely many primes ${\displaystyle p}$.

Let ${\displaystyle K}$ be a number field with ring of integers ${\displaystyle {\mathcal {O}}_{K}}$. At each finite place ${\displaystyle v}$, the completion ${\displaystyle K_{v}}$ is a finite extension of some ${\displaystyle \mathbb {Q} _{p}}$, and its valuation ring is denoted ${\displaystyle {\mathcal {O}}_{v}}$. At each infinite place, the completion is isomorphic to either ${\displaystyle \mathbb {R} }$ or ${\displaystyle \mathbb {C} }$. The adele ring is

${\displaystyle \mathbb {A} _{K}=\prod _{v\nmid \infty }'K_{v}\times \prod _{v\mid \infty }K_{v},}$

where the restricted product over finite places is taken with respect to the rings ${\displaystyle {\mathcal {O}}_{v}}$. Thus an adele of ${\displaystyle K}$ is a family ${\displaystyle (x_{v})_{v}}$ with ${\displaystyle x_{v}\in K_{v}}$ for every place ${\displaystyle v}$, such that ${\displaystyle x_{v}\in {\mathcal {O}}_{v}}$ for all but finitely many finite places.

For example, if ${\displaystyle K}$ is a quadratic number field, then its archimedean factor is either ${\displaystyle \mathbb {R} ^{2}}$, when ${\displaystyle K}$ has two real embeddings, or ${\displaystyle \mathbb {C} }$, when ${\displaystyle K}$ has one pair of complex embeddings. The finite part is a restricted product over the nonzero prime ideals of ${\displaystyle {\mathcal {O}}_{K}}$.²¹

If ${\displaystyle L/K}$ is a finite extension of number fields, then the adelic construction is compatible with extension of scalars. In particular, one has a natural isomorphism

${\displaystyle \mathbb {A} _{L}\cong \mathbb {A} _{K}\otimes _{K}L,}$

and, in the special case ${\displaystyle K=\mathbb {Q} }$,

${\displaystyle \mathbb {A} _{L}\cong \mathbb {A} _{\mathbb {Q} }\otimes _{\mathbb {Q} }L.}$

This gives another way to view the adele ring of a number field as the adelic extension of the rational adele ring.³¹

Now take the function field

${\displaystyle K=\mathbb {F} _{q}(\mathbb {P} ^{1})=\mathbb {F} _{q}(t)}$

of the projective line over a finite field. Its places correspond to the closed points ${\displaystyle x}$ of ${\displaystyle X=\mathbb {P} ^{1}}$. Such points may be described as maps

${\displaystyle x:\operatorname {Spec} \mathbb {F} _{q^{n}}\longrightarrow \mathbb {P} ^{1}}$

over ${\displaystyle \operatorname {Spec} \mathbb {F} _{q}}$. For instance, there are ${\displaystyle q+1}$ points of the form

${\displaystyle \operatorname {Spec} \mathbb {F} _{q}\longrightarrow \mathbb {P} ^{1}.}$

For a point ${\displaystyle x}$, the local ring used in the restricted product is the completed local ring

${\displaystyle {\widehat {\mathcal {O}}}_{X,x},}$

and the corresponding local field is its fraction field, often denoted ${\displaystyle K_{X,x}}$. Thus the adele ring of ${\displaystyle \mathbb {F} _{q}(\mathbb {P} ^{1})}$ may be written

${\displaystyle \mathbb {A} _{\mathbb {F} _{q}(\mathbb {P} ^{1})}=\prod _{x\in X}'K_{X,x},}$

where the restricted product is taken with respect to the completed local rings ${\displaystyle {\widehat {\mathcal {O}}}_{X,x}}$. Equivalently, its elements are families ${\displaystyle (f_{x})_{x}}$, with ${\displaystyle f_{x}\in K_{X,x}}$, such that ${\displaystyle f_{x}\in {\widehat {\mathcal {O}}}_{X,x}}$ for all but finitely many points ${\displaystyle x}$.¹

The same description holds for any smooth proper curve ${\displaystyle X/\mathbb {F} _{q}}$ over a finite field. If ${\displaystyle K=\mathbb {F} _{q}(X)}$ is its function field, then

${\displaystyle \mathbb {A} _{K}=\prod _{x\in X}'K_{X,x},}$

where ${\displaystyle x}$ runs over the closed points of ${\displaystyle X}$. Unlike number fields, global function fields have no archimedean places, so the finite adele ring and the full adele ring are the same.

The topology on the adele ring is the restricted product topology. For a finite set of places ${\displaystyle P}$ containing the archimedean places, define

${\displaystyle \mathbb {A} _{K}(P):=\prod _{v\in P}K_{v}\times \prod _{v\notin P}{\mathcal {O}}_{v}.}$

Equipped with the product topology and componentwise addition and multiplication, ${\displaystyle \mathbb {A} _{K}(P)}$ is a locally compact topological ring. If ${\displaystyle P'}$ is another finite set of places of ${\displaystyle K}$ containing ${\displaystyle P}$, then ${\displaystyle \mathbb {A} _{K}(P)}$ is an open subring of ${\displaystyle \mathbb {A} _{K}(P')}$. The adele ring is the union of all these open subrings:

${\displaystyle \mathbb {A} _{K}=\bigcup _{P\supset P_{\infty },\ |P|<\infty }\mathbb {A} _{K}(P).}$

Equivalently, ${\displaystyle \mathbb {A} _{K}}$ is the set of all ${\displaystyle x=(x_{v})_{v}}$ such that ${\displaystyle |x_{v}|_{v}\leq 1}$ for almost all non-archimedean places ${\displaystyle v}$. The topology of ${\displaystyle \mathbb {A} _{K}}$ is induced by the requirement that all ${\displaystyle \mathbb {A} _{K}(P)}$ be open subrings. Thus ${\displaystyle \mathbb {A} _{K}}$ is a locally compact topological ring.¹³

The same construction applies to sets of places. For every set of places ${\displaystyle S}$, the ring of ${\displaystyle S}$-adeles

${\displaystyle \mathbb {A} _{K,S}=\prod _{v\in S}'K_{v}}$

is a locally compact topological ring, with the restricted product topology. If

${\displaystyle \mathbb {A} _{K}^{S}=\prod _{v\notin S}'K_{v},}$

then there is a natural product decomposition

${\displaystyle \mathbb {A} _{K}\cong \mathbb {A} _{K,S}\times \mathbb {A} _{K}^{S}.}$

The diagonal embedding

${\displaystyle K\hookrightarrow \mathbb {A} _{K},\qquad a\mapsto (a,a,\ldots )}$

identifies ${\displaystyle K}$ with a subring of ${\displaystyle \mathbb {A} _{K}}$. With this embedding, the elements of ${\displaystyle K}$ are called principal adeles. The image of ${\displaystyle K}$ is discrete in ${\displaystyle \mathbb {A} _{K}}$, and the quotient

${\displaystyle \mathbb {A} _{K}/K}$

is compact. In particular, ${\displaystyle K}$ is closed in ${\displaystyle \mathbb {A} _{K}}$. This compactness property is one of the main reasons the adele ring is useful in harmonic analysis and in arithmetic applications.³¹

The adele ring also separates naturally into any chosen local factor and the remaining factors. Fix a place ${\displaystyle v}$ of ${\displaystyle K}$. Let ${\displaystyle P}$ be a finite set of places containing ${\displaystyle v}$ and ${\displaystyle P_{\infty }}$, and define

${\displaystyle \mathbb {A} _{K}'(P,v):=\prod _{w\in P\setminus \{v\}}K_{w}\times \prod _{w\notin P}{\mathcal {O}}_{w}.}$

Then

${\displaystyle \mathbb {A} _{K}(P)\cong K_{v}\times \mathbb {A} _{K}'(P,v).}$

Furthermore, define

${\displaystyle \mathbb {A} _{K}'(v):=\bigcup _{P\supset P_{\infty }\cup \{v\}}\mathbb {A} _{K}'(P,v),}$

where ${\displaystyle P}$ runs through all finite sets containing ${\displaystyle P_{\infty }\cup \{v\}}$. Then

${\displaystyle \mathbb {A} _{K}\cong K_{v}\times \mathbb {A} _{K}'(v),}$

via the map

${\displaystyle (a_{w})_{w}\mapsto (a_{v},(a_{w})_{w\neq v}).}$

Thus there is a natural embedding ${\displaystyle K_{v}\hookrightarrow \mathbb {A} _{K}}$ and a natural projection ${\displaystyle \mathbb {A} _{K}\twoheadrightarrow K_{v}}$. The same construction works with any finite set of places in place of the single place ${\displaystyle v}$.

Since ${\displaystyle \mathbb {A} _{K}}$ is locally compact as an additive group, it has an additive Haar measure. This measure is used in harmonic analysis on global fields and is usually normalized as a product of local Haar measures. With the standard normalization at the non-archimedean places, the valuation ring ${\displaystyle {\mathcal {O}}_{v}}$ has measure ${\displaystyle 1}$ for almost all finite places.⁵⁴

Since ${\displaystyle \mathbb {A} _{K}}$ is locally compact as an additive group, it has an additive Haar measure, usually denoted ${\displaystyle dx}$. This measure may be normalized as a product of local Haar measures on the completions ${\displaystyle K_{v}}$. At a non-archimedean place ${\displaystyle v}$, the local measure ${\displaystyle dx_{v}}$ is commonly normalized so that the valuation ring ${\displaystyle {\mathcal {O}}_{v}}$ has measure ${\displaystyle 1}$; at the archimedean places one uses the usual Lebesgue measure on ${\displaystyle \mathbb {R} }$ or ${\displaystyle \mathbb {C} }$.¹⁴

A function ${\displaystyle f:\mathbb {A} _{K}\to \mathbb {C} }$ is called simple if

${\displaystyle f=\prod _{v}f_{v},}$

where each ${\displaystyle f_{v}:K_{v}\to \mathbb {C} }$ is measurable and ${\displaystyle f_{v}=\mathbf {1} _{{\mathcal {O}}_{v}}}$ for almost all non-archimedean places ${\displaystyle v}$. With the standard normalization, every integrable simple function satisfies

${\displaystyle \int _{\mathbb {A} _{K}}f\,dx=\prod _{v}\int _{K_{v}}f_{v}\,dx_{v}.}$

The product is finite in the sense that almost all factors are equal to ${\displaystyle 1}$.⁷¹

Fourier analysis on the adele ring is based on the characters of its additive group. If ${\displaystyle G}$ is a locally compact abelian group, its character group ${\displaystyle {\widehat {G}}}$ is the group of all continuous homomorphisms from ${\displaystyle G}$ to

${\displaystyle \mathbb {T} =\{z\in \mathbb {C} :|z|=1\},}$

with the topology of uniform convergence on compact subsets. The adele ring is self-dual as a locally compact abelian group:

${\displaystyle \mathbb {A} _{K}\cong {\widehat {\mathbb {A} _{K}}}.}$

This is proved by reducing to the corresponding local statement for each completion ${\displaystyle K_{v}}$. For example, the usual character

${\displaystyle e_{\infty }(t)=\exp(2\pi it)}$

gives an isomorphism

${\displaystyle \mathbb {R} \longrightarrow {\widehat {\mathbb {R} }},\qquad s\mapsto {\bigl (}t\mapsto e_{\infty }(ts){\bigr )}.}$

Analogous local characters are used at the non-archimedean places, and their restricted product gives the global self-duality of ${\displaystyle \mathbb {A} _{K}}$.¹⁴

After choosing a nontrivial additive character ${\displaystyle \chi :\mathbb {A} _{K}\to \mathbb {T} }$, the Fourier transform of a suitable function ${\displaystyle f}$ on ${\displaystyle \mathbb {A} _{K}}$ is defined by

${\displaystyle {\widehat {f}}(y)=\int _{\mathbb {A} _{K}}f(x)\chi (xy)\,dx.}$

With a compatible choice of Haar measure, this Fourier transform satisfies the usual inversion and Plancherel formulas. One of the important features of the adelic setting is that global Fourier analysis factors into local Fourier analysis over the completions ${\displaystyle K_{v}}$.

With the help of the characters of ${\displaystyle \mathbb {A} _{K}}$, Fourier analysis can be done on the adele ring. In Tate's thesis, John Tate used Fourier analysis on the adele ring and integration over the idele group to study the Riemann zeta function, Dirichlet ${\displaystyle L}$-functions, and more general Hecke ${\displaystyle L}$-functions. Adelic forms of these functions can be represented as integrals over the adele ring or the idele group, with respect to corresponding Haar measures. Their functional equations and meromorphic continuations can then be proved by applying Fourier analysis and Poisson summation in the adelic setting.⁵⁶⁸

For example, for ${\displaystyle \operatorname {Re} (s)>1}$,

${\displaystyle \int _{\widehat {\mathbb {Z} }}|x|^{s}\,d^{\times }x=\zeta (s),}$

where ${\displaystyle d^{\times }x}$ is the multiplicative Haar measure on the finite idele group ${\displaystyle I_{\mathbb {Q} ,\mathrm {fin} }}$, normalized so that ${\displaystyle {\widehat {\mathbb {Z} }}^{\times }}$ has volume ${\displaystyle 1}$, and extended by zero to the finite adele ring. Thus the Riemann zeta function can be written as an integral over a subset of the adele ring.⁹

The adele ring enters class field theory through its group of units, the idele group ${\displaystyle \mathbb {A} _{K}^{\times }}$. The quotient

${\displaystyle C_{K}=\mathbb {A} _{K}^{\times }/K^{\times }}$

is the idele class group of ${\displaystyle K}$. Global class field theory describes the abelian extensions of ${\displaystyle K}$ in terms of topological quotients of ${\displaystyle C_{K}}$. In one formulation, the global Artin reciprocity law gives a reciprocity homomorphism from the idele class group to the Galois group of the maximal abelian extension of ${\displaystyle K}$. At finite level, for a finite abelian extension ${\displaystyle L/K}$, the corresponding quotient of ${\displaystyle C_{K}}$ is described using the norm subgroup from ${\displaystyle C_{L}}$.¹⁰¹¹

This adelic formulation packages the local reciprocity maps of local class field theory into a global statement. It replaces the older ideal-theoretic formulation, involving ideal class groups and ray class groups, by a statement about the topology and quotients of the idele class group.

The idele group gives a topological refinement of the group of fractional ideals of a number field. For a number field ${\displaystyle K}$, the finite part of the idele group maps onto the group of fractional ideals by

${\displaystyle (x_{\mathfrak {p}})_{\mathfrak {p}}\longmapsto \prod _{\mathfrak {p}}{\mathfrak {p}}^{v_{\mathfrak {p}}(x_{\mathfrak {p}})}.}$

The kernel is the product of the local unit groups. Consequently, the ordinary ideal class group can be recovered as a quotient of the idele class group. This viewpoint gives an adelic interpretation of the finiteness of the class number: the compactness of the norm-one idele classes implies that the ideal class group is compact, and since it is discrete, it is finite.³¹

The same circle of ideas also gives an adelic formulation of the unit theorem. If ${\displaystyle P}$ is a finite set of places containing the archimedean places, the group of ${\displaystyle P}$-units appears as an intersection of ${\displaystyle K^{\times }}$ with a natural open subgroup of the idele group. In particular, for a number field ${\displaystyle K}$, Dirichlet's unit theorem states that

${\displaystyle {\mathcal {O}}_{K}^{\times }\cong \mu (K)\times \mathbb {Z} ^{r+s-1},}$

where ${\displaystyle \mu (K)}$ is the finite cyclic group of roots of unity in ${\displaystyle K}$, ${\displaystyle r}$ is the number of real embeddings, and ${\displaystyle s}$ is the number of conjugate pairs of complex embeddings.¹³

The topology on ${\displaystyle \mathbb {A} _{K}}$ makes the quotient ${\displaystyle \mathbb {A} _{K}/K}$ compact, allowing one to do harmonic analysis on the adele ring. With the help of the characters of ${\displaystyle \mathbb {A} _{K}}$, Fourier analysis can be done on the adele ring; integration over the idele group then gives zeta integrals.⁵⁴

In Tate's thesis, John Tate used Fourier analysis on the adele ring and the idele group to study the Riemann zeta function, Dirichlet ${\displaystyle L}$-functions, and more general Hecke ${\displaystyle L}$-functions. Adelic forms of these functions can be represented as integrals over the adele ring or the idele group, with respect to corresponding Haar measures. Their functional equations and meromorphic continuations can then be proved using Fourier analysis and Poisson summation in the adelic setting.⁵¹²⁶

For example, for ${\displaystyle \operatorname {Re} (s)>1}$, one has an adelic integral representation of the Riemann zeta function,

${\displaystyle \int _{\widehat {\mathbb {Z} }}|x|^{s}\,d^{\times }x=\zeta (s),}$

where ${\displaystyle d^{\times }x}$ is the multiplicative Haar measure on the finite idele group ${\displaystyle I_{\mathbb {Q} ,\mathrm {fin} }}$, normalized so that ${\displaystyle {\widehat {\mathbb {Z} }}^{\times }}$ has volume ${\displaystyle 1}$, and extended by zero to the finite adele ring. Thus the Riemann zeta function can be written as an integral over a subset of the adele ring.

Adeles also provide the natural language for automorphic forms. Instead of studying functions separately over the real, complex, and ${\displaystyle p}$-adic points of an algebraic group, one studies functions on adelic groups such as ${\displaystyle G(\mathbb {A} _{K})}$. For example, automorphic forms for ${\displaystyle \operatorname {GL} _{2}}$ over ${\displaystyle \mathbb {Q} }$ may be viewed as functions on

${\displaystyle \operatorname {GL} _{2}(\mathbb {Q} )\backslash \operatorname {GL} _{2}(\mathbb {A} _{\mathbb {Q} })}$

satisfying suitable algebraic, analytic, and growth conditions. In this setting, automorphic ${\displaystyle L}$-functions can often be described by integrals over adelic groups.¹³¹⁴

More generally, the use of adelic points ${\displaystyle G(\mathbb {A} _{K})}$ for reductive algebraic groups ${\displaystyle G}$ is central in the modern theory of automorphic representations. This viewpoint is also one of the starting points of the Langlands program, which relates automorphic representations of adelic groups to Galois representations.¹³

The adele ring provides a unified interpretation of approximation theorems and local-global questions. The weak approximation theorem says that, for finitely many inequivalent valuations of ${\displaystyle K}$, the diagonal image of ${\displaystyle K}$ is dense in the product of the corresponding completions. The strong approximation theorem says that, after omitting one place ${\displaystyle v_{0}}$, the field ${\displaystyle K}$ is dense in the restricted product over all other places. Thus the global field is discrete in its full adele ring, but becomes dense when one place is omitted.³

Adelic language is also used to formulate local-global principles, such as the Hasse principle. In such problems one compares solutions over the global field ${\displaystyle K}$ with compatible families of solutions over all completions ${\displaystyle K_{v}}$. The adele ring provides a single space in which these local conditions can be collected and studied together.

For a smooth proper curve ${\displaystyle X/\mathbb {F} _{q}}$ with function field ${\displaystyle K}$, the adele ring of ${\displaystyle K}$ can be described using the completions at the closed points of ${\displaystyle X}$. In this setting, ideles recover the divisor and Picard groups of the curve. One has

${\displaystyle \operatorname {Div} (X)=\mathbb {A} _{X}^{\times }/\mathbb {O} _{X}^{\times }}$

and

${\displaystyle \operatorname {Pic} (X)=K^{\times }\backslash \mathbb {A} _{X}^{\times }/\mathbb {O} _{X}^{\times }.}$

Thus the divisor-class description of line bundles on a curve can be expressed adelically.

More generally, for an algebraic group ${\displaystyle G}$, adelic double quotients describe moduli of bundles on curves. In Weil uniformization, for suitable groups such as semisimple groups, and also for ${\displaystyle \operatorname {GL} _{n}}$, one has an adelic description of the form

${\displaystyle \operatorname {Bun} _{G}(X)=G(K)\backslash G(\mathbb {A} _{X})/G(\mathbb {O} _{X}).}$

For ${\displaystyle G=\mathbb {G} _{m}}$, this recovers the adelic description of the Picard group.

Adeles also occur in the cohomology of algebraic curves. If ${\displaystyle X}$ is a smooth proper curve over the complex numbers, one can define the adeles of its function field ${\displaystyle \mathbb {C} (X)}$ in a way analogous to the function-field case over finite fields. Tate proved that Serre duality on ${\displaystyle X}$,

${\displaystyle H^{1}(X,{\mathcal {L}})\simeq H^{0}(X,\Omega _{X}\otimes {\mathcal {L}}^{-1})^{*},}$

can be deduced by working with this adele ring ${\displaystyle \mathbb {A} _{\mathbb {C} (X)}}$, where ${\displaystyle {\mathcal {L}}}$ is a line bundle on ${\displaystyle X}$.¹⁵

The idele group of a global field ${\displaystyle K}$ is the group of invertible elements of the adele ring ${\displaystyle \mathbb {A} _{K}}$. It is usually denoted

${\displaystyle \mathbb {A} _{K}^{\times }}$

or ${\displaystyle I_{K}}$. Equivalently, it is the restricted direct product

${\displaystyle \mathbb {A} _{K}^{\times }=\prod _{v}'K_{v}^{\times }}$

of the multiplicative groups of the completions ${\displaystyle K_{v}}$, taken with respect to the unit groups ${\displaystyle {\mathcal {O}}_{v}^{\times }}$ at the non-archimedean places. Thus an idele is a family ${\displaystyle x=(x_{v})_{v}}$, with ${\displaystyle x_{v}\in K_{v}^{\times }}$ for every place ${\displaystyle v}$, such that ${\displaystyle x_{v}\in {\mathcal {O}}_{v}^{\times }}$ for all but finitely many non-archimedean ${\displaystyle v}$.

Although ${\displaystyle \mathbb {A} _{K}^{\times }}$ is the group of units of the adele ring, it is not given the subspace topology inherited from ${\displaystyle \mathbb {A} _{K}}$. Instead it is given the restricted product topology, equivalently the topology induced by the embedding

${\displaystyle \mathbb {A} _{K}^{\times }\longrightarrow \mathbb {A} _{K}\times \mathbb {A} _{K},\qquad x\mapsto (x,x^{-1}).}$

With this topology, ${\displaystyle \mathbb {A} _{K}^{\times }}$ is an abelian locally compact topological group.

The diagonal embedding of ${\displaystyle K^{\times }}$ into ${\displaystyle \mathbb {A} _{K}^{\times }}$ gives the subgroup of principal ideles. The quotient

${\displaystyle C_{K}=\mathbb {A} _{K}^{\times }/K^{\times }}$

is the idele class group. This group is a central object in class field theory, where abelian extensions of ${\displaystyle K}$ are described in terms of topological quotients of ${\displaystyle C_{K}}$.

The idele group also carries a natural absolute value, or module,

${\displaystyle |x|_{\mathbb {A} }=\prod _{v}|x_{v}|_{v},}$

where the local absolute values are normalized in the standard way. The product is finite for ideles, since almost all finite components are units. The subgroup

${\displaystyle \mathbb {A} _{K}^{1}=\{x\in \mathbb {A} _{K}^{\times }:|x|_{\mathbb {A} }=1\}}$

is the group of norm-one ideles. By the product formula, ${\displaystyle K^{\times }}$ lies in ${\displaystyle \mathbb {A} _{K}^{1}}$, and the quotient ${\displaystyle \mathbb {A} _{K}^{1}/K^{\times }}$ is compact.

For number fields, the finite part of the idele group maps naturally onto the group of fractional ideals by

${\displaystyle (x_{\mathfrak {p}})_{\mathfrak {p}}\longmapsto \prod _{\mathfrak {p}}{\mathfrak {p}}^{v_{\mathfrak {p}}(x_{\mathfrak {p}})}.}$

The kernel is ${\displaystyle {\widehat {\mathcal {O}}}_{K}^{\times }}$, so the ordinary ideal class group is recovered as a quotient of the idele class group. In this way, the idele class group refines the ideal class group by retaining local unit data and archimedean information.

Ideles are also used in harmonic analysis on global fields. In Tate's thesis, integration over the adele ring and the idele group gives a uniform treatment of Hecke ${\displaystyle L}$-functions, including their Euler products, analytic continuation, and functional equations.

The preceding sections give the basic definition and main uses of the adele ring. This section records some standard structural facts and proof sketches.

The difference between the restricted and unrestricted product topologies can be illustrated using a sequence in ${\displaystyle \mathbb {A} _{\mathbb {Q} }}$.

Lemma. Consider the following sequence in ${\displaystyle \mathbb {A} _{\mathbb {Q} }}$:

${\displaystyle {\begin{aligned}x_{1}&=\left({\frac {1}{2}},1,1,\ldots \right)\\x_{2}&=\left(1,{\frac {1}{3}},1,\ldots \right)\\x_{3}&=\left(1,1,{\frac {1}{5}},1,\ldots \right)\\x_{4}&=\left(1,1,1,{\frac {1}{7}},1,\ldots \right)\\&\vdots \end{aligned}}}$

In the product topology this converges to ${\displaystyle (1,1,\ldots )}$, 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 does not converge in restricted product topology. For each adele ${\displaystyle a=(a_{p})_{p}\in \mathbb {A} _{\mathbb {Q} }}$ and for each restricted open rectangle ${\displaystyle \textstyle U=\prod _{p\in E}U_{p}\times \prod _{p\notin E}\mathbb {Z} _{p},}$ it has ${\displaystyle {\tfrac {1}{p}}-a_{p}\notin \mathbb {Z} _{p}}$ for ${\displaystyle a_{p}\in \mathbb {Z} _{p}}$ and therefore ${\displaystyle {\tfrac {1}{p}}-a_{p}\notin \mathbb {Z} _{p}}$ for all ${\displaystyle p\notin F.}$ As a result ${\displaystyle x_{n}-a\notin U}$ for almost all ${\displaystyle n\in \mathbb {N} .}$ In this consideration, ${\displaystyle E}$ and ${\displaystyle F}$ are finite subsets of the set of all places.

The profinite integers are defined as the profinite completion of the rings ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ with the partial order ${\displaystyle n\geq m\Leftrightarrow m|n,}$ i.e.,

${\displaystyle {\widehat {\mathbb {Z} }}:=\varprojlim _{n}\mathbb {Z} /n\mathbb {Z} .}$

Lemma. ${\displaystyle \textstyle {\widehat {\mathbb {Z} }}\cong \prod _{p}\mathbb {Z} _{p}.}$

Proof. This follows from the Chinese Remainder Theorem.

Lemma. ${\displaystyle \mathbb {A} _{\mathbb {Q} ,\mathrm {fin} }={\widehat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q} .}$

Proof. Use the universal property of the tensor product. Define a ${\displaystyle \mathbb {Z} }$-bilinear function

${\displaystyle {\begin{cases}\Psi :{\widehat {\mathbb {Z} }}\times \mathbb {Q} \to \mathbb {A} _{\mathbb {Q} ,\mathrm {fin} }\\\left((a_{p})_{p},q\right)\mapsto (a_{p}q)_{p}.\end{cases}}}$

This is well-defined because for a given ${\displaystyle q={\tfrac {m}{n}}\in \mathbb {Q} }$ with ${\displaystyle m,n}$ coprime there are only finitely many primes dividing ${\displaystyle n.}$ Let ${\displaystyle M}$ be another ${\displaystyle \mathbb {Z} }$-module with a ${\displaystyle \mathbb {Z} }$-bilinear map ${\displaystyle \Phi :{\widehat {\mathbb {Z} }}\times \mathbb {Q} \to M.}$ It must be the case that ${\displaystyle \Phi }$ factors through ${\displaystyle \Psi }$ uniquely, i.e., there exists a unique ${\displaystyle \mathbb {Z} }$-linear map ${\displaystyle {\tilde {\Phi }}:\mathbb {A} _{\mathbb {Q} ,\mathrm {fin} }\to M}$ such that ${\displaystyle \Phi ={\tilde {\Phi }}\circ \Psi .}$ ${\displaystyle {\tilde {\Phi }}}$ can be defined as follows: for a given ${\displaystyle (u_{p})_{p}}$ there exist ${\displaystyle u\in \mathbb {N} }$ and ${\displaystyle (v_{p})_{p}\in {\widehat {\mathbb {Z} }}}$ such that ${\displaystyle u_{p}={\tfrac {1}{u}}\cdot v_{p}}$ for all ${\displaystyle p.}$ Define ${\displaystyle {\tilde {\Phi }}((u_{p})_{p}):=\Phi ((v_{p})_{p},{\tfrac {1}{u}}).}$ One can show ${\displaystyle {\tilde {\Phi }}}$ is well-defined, ${\displaystyle \mathbb {Z} }$-linear, satisfies ${\displaystyle \Phi ={\tilde {\Phi }}\circ \Psi }$ and is unique with these properties.

Corollary. Define ${\displaystyle \mathbb {A} _{\mathbb {Z} }:={\widehat {\mathbb {Z} }}\times \mathbb {R} .}$ This results in an algebraic isomorphism ${\displaystyle \mathbb {A} _{\mathbb {Q} }\cong \mathbb {A} _{\mathbb {Z} }\otimes _{\mathbb {Z} }\mathbb {Q} .}$

Proof.

${\displaystyle \mathbb {A} _{\mathbb {Z} }\otimes _{\mathbb {Z} }\mathbb {Q} =\left({\widehat {\mathbb {Z} }}\times \mathbb {R} \right)\otimes _{\mathbb {Z} }\mathbb {Q} \cong \left({\widehat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q} \right)\times (\mathbb {R} \otimes _{\mathbb {Z} }\mathbb {Q} )\cong \left({\widehat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q} \right)\times \mathbb {R} =\mathbb {A} _{\mathbb {Q} ,\mathrm {fin} }\times \mathbb {R} =\mathbb {A} _{\mathbb {Q} }.}$

Lemma. For a number field ${\displaystyle K}$, ${\displaystyle \mathbb {A} _{K}=\mathbb {A} _{\mathbb {Q} }\otimes _{\mathbb {Q} }K.}$

Remark. Using ${\displaystyle \mathbb {A} _{\mathbb {Q} }\otimes _{\mathbb {Q} }K\cong \mathbb {A} _{\mathbb {Q} }\oplus \dots \oplus \mathbb {A} _{\mathbb {Q} },}$ where there are ${\displaystyle [K:\mathbb {Q} ]}$ summands, the right side receives the product topology and this topology is transported via the isomorphism onto ${\displaystyle \mathbb {A} _{\mathbb {Q} }\otimes _{\mathbb {Q} }K.}$

If ${\displaystyle L/K}$ is a finite extension, then ${\displaystyle L}$ is a global field. Thus ${\displaystyle \mathbb {A} _{L}}$ is defined, and ${\displaystyle \textstyle \mathbb {A} _{L}={\prod _{v}}'L_{v}.}$ The ring ${\displaystyle \mathbb {A} _{K}}$ can be identified with a subring of ${\displaystyle \mathbb {A} _{L}.}$ Map ${\displaystyle a=(a_{v})_{v}\in \mathbb {A} _{K}}$ to ${\displaystyle a'=(a'_{w})_{w}\in \mathbb {A} _{L}}$, where ${\displaystyle a'_{w}=a_{v}\in K_{v}\subset L_{w}}$ for ${\displaystyle w|v.}$ Then ${\displaystyle a=(a_{w})_{w}\in \mathbb {A} _{L}}$ is in the subring ${\displaystyle \mathbb {A} _{K}}$ if ${\displaystyle a_{w}\in K_{v}}$ for ${\displaystyle w|v}$ and ${\displaystyle a_{w}=a_{w'}}$ for all ${\displaystyle w,w'}$ lying above the same place ${\displaystyle v}$ of ${\displaystyle K.}$

Lemma. If ${\displaystyle L/K}$ is a finite extension, then ${\displaystyle \mathbb {A} _{L}\cong \mathbb {A} _{K}\otimes _{K}L}$ both algebraically and topologically.

With the help of this isomorphism, the inclusion ${\displaystyle \mathbb {A} _{K}\subset \mathbb {A} _{L}}$ is given by

${\displaystyle {\begin{cases}\mathbb {A} _{K}\to \mathbb {A} _{L}\\\alpha \mapsto \alpha \otimes _{K}1.\end{cases}}}$

Furthermore, the principal adeles in ${\displaystyle \mathbb {A} _{K}}$ can be identified with a subgroup of principal adeles in ${\displaystyle \mathbb {A} _{L}}$ via the natural embedding ${\displaystyle K\to L.}$

Proof.¹⁶ Let ${\displaystyle \omega _{1},\ldots ,\omega _{n}}$ be a basis of ${\displaystyle L}$ over ${\displaystyle K.}$ Then for almost all ${\displaystyle v,}$

${\displaystyle {\widetilde {O_{v}}}\cong O_{v}\omega _{1}\oplus \cdots \oplus O_{v}\omega _{n}.}$

Furthermore, there are the following isomorphisms:

${\displaystyle K_{v}\omega _{1}\oplus \cdots \oplus K_{v}\omega _{n}\cong K_{v}\otimes _{K}L\cong L_{v}=\prod \nolimits _{w|v}L_{w}.}$

For the second use the map

${\displaystyle {\begin{cases}K_{v}\otimes _{K}L\to L_{v}\\\alpha _{v}\otimes a\mapsto (\alpha _{v}\cdot (\tau _{w}(a)))_{w}\end{cases}}}$

in which ${\displaystyle \tau _{w}:L\to L_{w}}$ is the canonical embedding and ${\displaystyle w|v.}$ The restricted product is taken on both sides with respect to ${\displaystyle {\widetilde {O_{v}}}:}$

${\displaystyle {\begin{aligned}\mathbb {A} _{K}\otimes _{K}L&=\left({\prod _{v}}'K_{v}\right)\otimes _{K}L\\&\cong {\prod _{v}}'(K_{v}\omega _{1}\oplus \cdots \oplus K_{v}\omega _{n})\\&\cong {\prod _{v}}'(K_{v}\otimes _{K}L)\\&\cong {\prod _{v}}'L_{v}\\&=\mathbb {A} _{L}.\end{aligned}}}$

Corollary. As additive groups ${\displaystyle \mathbb {A} _{L}\cong \mathbb {A} _{K}\oplus \cdots \oplus \mathbb {A} _{K},}$ where the right side has ${\displaystyle [L:K]}$ summands.

The set of principal adeles in ${\displaystyle \mathbb {A} _{L}}$ is identified with the set ${\displaystyle K\oplus \cdots \oplus K}$, where the left side has ${\displaystyle [L:K]}$ summands and ${\displaystyle K}$ is considered as a subset of ${\displaystyle \mathbb {A} _{K}.}$

Let ${\displaystyle E}$ be a finite-dimensional vector space over ${\displaystyle K}$ and ${\displaystyle \{\omega _{1},\ldots ,\omega _{n}\}}$ a basis for ${\displaystyle E}$ over ${\displaystyle K.}$ For each place ${\displaystyle v}$ of ${\displaystyle K}$:

${\displaystyle {\begin{aligned}E_{v}&:=E\otimes _{K}K_{v}\cong K_{v}\omega _{1}\oplus \cdots \oplus K_{v}\omega _{n},\\{\widetilde {O_{v}}}&:=O_{v}\omega _{1}\oplus \cdots \oplus O_{v}\omega _{n}.\end{aligned}}}$

The adele ring of ${\displaystyle E}$ is defined as

${\displaystyle \mathbb {A} _{E}:={\prod _{v}}'E_{v}.}$

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 the adele ring for number fields. Next, ${\displaystyle \mathbb {A} _{E}}$ is equipped with the restricted product topology. Then ${\displaystyle \mathbb {A} _{E}=E\otimes _{K}\mathbb {A} _{K}}$ and ${\displaystyle E}$ is embedded in ${\displaystyle \mathbb {A} _{E}}$ naturally via the map ${\displaystyle e\mapsto e\otimes 1.}$

An alternative definition of the topology on ${\displaystyle \mathbb {A} _{E}}$ can be provided. Consider all linear maps ${\displaystyle E\to K.}$ Using the natural embeddings ${\displaystyle E\to \mathbb {A} _{E}}$ and ${\displaystyle K\to \mathbb {A} _{K},}$ extend these linear maps to ${\displaystyle \mathbb {A} _{E}\to \mathbb {A} _{K}.}$ The topology on ${\displaystyle \mathbb {A} _{E}}$ is the coarsest topology for which all these extensions are continuous.

The topology can be defined in a different way. Fixing a basis for ${\displaystyle E}$ over ${\displaystyle K}$ results in an isomorphism ${\displaystyle E\cong K^{n}.}$ Therefore fixing a basis induces an isomorphism ${\displaystyle (\mathbb {A} _{K})^{n}\cong \mathbb {A} _{E}.}$ The left-hand side is supplied with the product topology and this topology is transported with the isomorphism onto the right-hand side. The topology does not 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,

${\displaystyle {\begin{aligned}\mathbb {A} _{E}&=E\otimes _{K}\mathbb {A} _{K}\\&\cong (K\otimes _{K}\mathbb {A} _{K})\oplus \cdots \oplus (K\otimes _{K}\mathbb {A} _{K})\\&\cong \mathbb {A} _{K}\oplus \cdots \oplus \mathbb {A} _{K},\end{aligned}}}$

where the sums have ${\displaystyle n}$ summands. In case of ${\displaystyle E=L,}$ the definition above is consistent with the results about the adele ring of a finite extension ${\displaystyle L/K.}$¹⁷

Let ${\displaystyle A}$ be a finite-dimensional algebra over ${\displaystyle K.}$ In particular, ${\displaystyle A}$ is a finite-dimensional vector space over ${\displaystyle K.}$ As a consequence, ${\displaystyle \mathbb {A} _{A}}$ is defined and ${\displaystyle \mathbb {A} _{A}\cong \mathbb {A} _{K}\otimes _{K}A.}$ Since there is multiplication on ${\displaystyle \mathbb {A} _{K}}$ and ${\displaystyle A,}$ a multiplication on ${\displaystyle \mathbb {A} _{A}}$ can be defined via

${\displaystyle \forall \alpha ,\beta \in \mathbb {A} _{K}{\text{ and }}\forall a,b\in A:\qquad (\alpha \otimes _{K}a)\cdot (\beta \otimes _{K}b):=(\alpha \beta )\otimes _{K}(ab).}$

As a consequence, ${\displaystyle \mathbb {A} _{A}}$ is an algebra with a unit over ${\displaystyle \mathbb {A} _{K}.}$ Let ${\displaystyle {\mathcal {B}}}$ be a finite subset of ${\displaystyle A}$, containing a basis for ${\displaystyle A}$ over ${\displaystyle K.}$ For any finite place ${\displaystyle v}$, ${\displaystyle M_{v}}$ is defined as the ${\displaystyle O_{v}}$-module generated by ${\displaystyle {\mathcal {B}}}$ in ${\displaystyle A_{v}.}$ For each finite set of places ${\displaystyle P\supset P_{\infty },}$ define

${\displaystyle \mathbb {A} _{A}(P,\alpha )=\prod _{v\in P}A_{v}\times \prod _{v\notin P}M_{v}.}$

One can show there is a finite set ${\displaystyle P_{0}}$ so that ${\displaystyle \mathbb {A} _{A}(P,\alpha )}$ is an open subring of ${\displaystyle \mathbb {A} _{A}}$, if ${\displaystyle P\supset P_{0}.}$ Furthermore ${\displaystyle \mathbb {A} _{A}}$ is the union of all these subrings and for ${\displaystyle A=K}$, the definition above is consistent with the definition of the adele ring.

Let ${\displaystyle L/K}$ be a finite extension. Since ${\displaystyle \mathbb {A} _{K}=\mathbb {A} _{K}\otimes _{K}K}$ and ${\displaystyle \mathbb {A} _{L}=\mathbb {A} _{K}\otimes _{K}L}$ from the lemma above, ${\displaystyle \mathbb {A} _{K}}$ can be interpreted as a closed subring of ${\displaystyle \mathbb {A} _{L}.}$ For this embedding, write ${\displaystyle \operatorname {con} _{L/K}}$. Explicitly, for all places ${\displaystyle w}$ of ${\displaystyle L}$ above ${\displaystyle v}$ and for any ${\displaystyle \alpha \in \mathbb {A} _{K}}$,

${\displaystyle (\operatorname {con} _{L/K}(\alpha ))_{w}=\alpha _{v}\in K_{v}.}$

Let ${\displaystyle M/L/K}$ be a tower of global fields. Then

${\displaystyle \operatorname {con} _{M/K}(\alpha )=\operatorname {con} _{M/L}(\operatorname {con} _{L/K}(\alpha ))\qquad \forall \alpha \in \mathbb {A} _{K}.}$

Furthermore, restricted to the principal adeles ${\displaystyle \operatorname {con} }$ is the natural injection ${\displaystyle K\to L.}$

Let ${\displaystyle \{\omega _{1},\ldots ,\omega _{n}\}}$ be a basis of the field extension ${\displaystyle L/K.}$ Then each ${\displaystyle \alpha \in \mathbb {A} _{L}}$ can be written as ${\displaystyle \textstyle \sum _{j=1}^{n}\alpha _{j}\omega _{j}}$, where ${\displaystyle \alpha _{j}\in \mathbb {A} _{K}}$ are unique. The map ${\displaystyle \alpha \mapsto \alpha _{j}}$ is continuous. Define ${\displaystyle \alpha _{ij}}$, depending on ${\displaystyle \alpha }$, via the equations

${\displaystyle {\begin{aligned}\alpha \omega _{1}&=\sum _{j=1}^{n}\alpha _{1j}\omega _{j},\\&\vdots \\\alpha \omega _{n}&=\sum _{j=1}^{n}\alpha _{nj}\omega _{j}.\end{aligned}}}$

Now define the trace and norm of ${\displaystyle \alpha }$ as

${\displaystyle {\begin{aligned}\operatorname {Tr} _{L/K}(\alpha )&:=\operatorname {Tr} ((\alpha _{ij})_{i,j})=\sum _{i=1}^{n}\alpha _{ii},\\N_{L/K}(\alpha )&:=N((\alpha _{ij})_{i,j})=\det((\alpha _{ij})_{i,j}).\end{aligned}}}$

These are the trace and the determinant of the linear map

${\displaystyle {\begin{cases}\mathbb {A} _{L}\to \mathbb {A} _{L}\\x\mapsto \alpha x.\end{cases}}}$

They are continuous maps on the adele ring, and they fulfil the usual equations:

${\displaystyle {\begin{aligned}\operatorname {Tr} _{L/K}(\alpha +\beta )&=\operatorname {Tr} _{L/K}(\alpha )+\operatorname {Tr} _{L/K}(\beta )&&\forall \alpha ,\beta \in \mathbb {A} _{L},\\\operatorname {Tr} _{L/K}(\operatorname {con} (\alpha ))&=n\alpha &&\forall \alpha \in \mathbb {A} _{K},\\N_{L/K}(\alpha \beta )&=N_{L/K}(\alpha )N_{L/K}(\beta )&&\forall \alpha ,\beta \in \mathbb {A} _{L},\\N_{L/K}(\operatorname {con} (\alpha ))&=\alpha ^{n}&&\forall \alpha \in \mathbb {A} _{K}.\end{aligned}}}$

Furthermore, for ${\displaystyle \alpha \in L}$, ${\displaystyle \operatorname {Tr} _{L/K}(\alpha )}$ and ${\displaystyle N_{L/K}(\alpha )}$ are identical to the trace and norm of the field extension ${\displaystyle L/K.}$ For a tower of fields ${\displaystyle M/L/K}$, the result is

${\displaystyle {\begin{aligned}\operatorname {Tr} _{L/K}(\operatorname {Tr} _{M/L}(\alpha ))&=\operatorname {Tr} _{M/K}(\alpha )&&\forall \alpha \in \mathbb {A} _{M},\\N_{L/K}(N_{M/L}(\alpha ))&=N_{M/K}(\alpha )&&\forall \alpha \in \mathbb {A} _{M}.\end{aligned}}}$

Moreover, it can be proven that:¹⁸

${\displaystyle {\begin{aligned}\operatorname {Tr} _{L/K}(\alpha )&=\left(\sum _{w|v}\operatorname {Tr} _{L_{w}/K_{v}}(\alpha _{w})\right)_{v}&&\forall \alpha \in \mathbb {A} _{L},\\N_{L/K}(\alpha )&=\left(\prod _{w|v}N_{L_{w}/K_{v}}(\alpha _{w})\right)_{v}&&\forall \alpha \in \mathbb {A} _{L}.\end{aligned}}}$

Theorem.¹⁹ ${\displaystyle K}$ is discrete and cocompact in ${\displaystyle \mathbb {A} _{K}.}$ In particular, ${\displaystyle K}$ is closed in ${\displaystyle \mathbb {A} _{K}.}$

Proof. Prove the case ${\displaystyle K=\mathbb {Q} .}$ To show ${\displaystyle \mathbb {Q} \subset \mathbb {A} _{\mathbb {Q} }}$ is discrete it is sufficient to show the existence of a neighbourhood of ${\displaystyle 0}$ which contains no other rational number. The general case follows via translation. Define

${\displaystyle U:=\left\{(\alpha _{p})_{p}:\forall p<\infty ,\ |\alpha _{p}|_{p}\leq 1\quad {\text{and}}\quad |\alpha _{\infty }|_{\infty }<1\right\}={\widehat {\mathbb {Z} }}\times (-1,1).}$

${\displaystyle U}$ is an open neighbourhood of ${\displaystyle 0\in \mathbb {A} _{\mathbb {Q} }.}$ It is claimed that ${\displaystyle U\cap \mathbb {Q} =\{0\}.}$ Let ${\displaystyle \beta \in U\cap \mathbb {Q} .}$ Then ${\displaystyle \beta \in \mathbb {Q} }$ and ${\displaystyle |\beta |_{p}\leq 1}$ for all ${\displaystyle p}$, and therefore ${\displaystyle \beta \in \mathbb {Z} .}$ Additionally, ${\displaystyle \beta \in (-1,1)}$ and therefore ${\displaystyle \beta =0.}$

Next, to show compactness, define

${\displaystyle W:=\left\{(\alpha _{p})_{p}:\forall p<\infty ,\ |\alpha _{p}|_{p}\leq 1\quad {\text{and}}\quad |\alpha _{\infty }|_{\infty }\leq {\frac {1}{2}}\right\}={\widehat {\mathbb {Z} }}\times \left[-{\frac {1}{2}},{\frac {1}{2}}\right].}$

Each element in ${\displaystyle \mathbb {A} _{\mathbb {Q} }/\mathbb {Q} }$ has a representative in ${\displaystyle W}$, that is, for each ${\displaystyle \alpha \in \mathbb {A} _{\mathbb {Q} }}$, there exists ${\displaystyle \beta \in \mathbb {Q} }$ such that ${\displaystyle \alpha -\beta \in W.}$ Let ${\displaystyle \alpha =(\alpha _{p})_{p}\in \mathbb {A} _{\mathbb {Q} }}$ be arbitrary and ${\displaystyle p}$ be a prime for which ${\displaystyle |\alpha _{p}|>1.}$ Then there exists ${\displaystyle r_{p}=z_{p}/p^{x_{p}}}$, with ${\displaystyle z_{p}\in \mathbb {Z} }$ and ${\displaystyle x_{p}\in \mathbb {N} }$, such that ${\displaystyle |\alpha _{p}-r_{p}|\leq 1.}$ Replace ${\displaystyle \alpha }$ with ${\displaystyle \alpha -r_{p}}$ and let ${\displaystyle q\neq p}$ be another prime. Then

${\displaystyle \left|\alpha _{q}-r_{p}\right|_{q}\leq \max \left\{|\alpha _{q}|_{q},|r_{p}|_{q}\right\}\leq \max \left\{|\alpha _{q}|_{q},1\right\}\leq 1.}$

Next, it can be claimed that

${\displaystyle |\alpha _{q}-r_{p}|_{q}\leq 1\Longleftrightarrow |\alpha _{q}|_{q}\leq 1.}$

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 ${\displaystyle \alpha }$ are not in ${\displaystyle \mathbb {Z} _{p}}$ is reduced by one. With iteration, it can be deduced that there exists ${\displaystyle r\in \mathbb {Q} }$ such that ${\displaystyle \alpha -r\in {\widehat {\mathbb {Z} }}\times \mathbb {R} .}$ Now select ${\displaystyle s\in \mathbb {Z} }$ such that ${\displaystyle \alpha _{\infty }-r-s\in [-{\tfrac {1}{2}},{\tfrac {1}{2}}].}$ Then ${\displaystyle \alpha -(r+s)\in W.}$ The continuous projection ${\displaystyle \pi :W\to \mathbb {A} _{\mathbb {Q} }/\mathbb {Q} }$ is surjective, therefore ${\displaystyle \mathbb {A} _{\mathbb {Q} }/\mathbb {Q} }$, as the continuous image of a compact set, is compact.

Corollary. Let ${\displaystyle E}$ be a finite-dimensional vector space over ${\displaystyle K.}$ Then ${\displaystyle E}$ is discrete and cocompact in ${\displaystyle \mathbb {A} _{E}.}$

Weak Approximation Theorem.²⁰ Let ${\displaystyle |\cdot |_{1},\ldots ,|\cdot |_{N}}$ be inequivalent valuations of ${\displaystyle K.}$ Let ${\displaystyle K_{n}}$ be the completion of ${\displaystyle K}$ with respect to ${\displaystyle |\cdot |_{n}.}$ Embed ${\displaystyle K}$ diagonally in ${\displaystyle K_{1}\times \cdots \times K_{N}.}$ Then ${\displaystyle K}$ is everywhere dense in ${\displaystyle K_{1}\times \cdots \times K_{N}.}$ In other words, for each ${\displaystyle \varepsilon >0}$ and for each ${\displaystyle (\alpha _{1},\ldots ,\alpha _{N})\in K_{1}\times \cdots \times K_{N}}$, there exists ${\displaystyle \xi \in K}$ such that

${\displaystyle \forall n\in \{1,\ldots ,N\}:\quad |\alpha _{n}-\xi |_{n}<\varepsilon .}$

Strong Approximation Theorem.²¹ Let ${\displaystyle v_{0}}$ be a place of ${\displaystyle K.}$ Define

${\displaystyle V:={\prod _{v\neq v_{0}}}'K_{v}.}$

Then ${\displaystyle K}$ is dense in ${\displaystyle V.}$

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 ${\displaystyle K}$ is turned into a denseness of ${\displaystyle K.}$

Theorem (finiteness of the class number of a number field). Let ${\displaystyle K}$ be a number field. Then ${\displaystyle |\operatorname {Cl} _{K}|<\infty .}$

Proof. The map

${\displaystyle {\begin{cases}I_{K}^{1}\to J_{K}\\\left((\alpha _{v})_{v<\infty },(\alpha _{v})_{v|\infty }\right)\mapsto \prod _{v<\infty }{\mathfrak {p}}_{v}^{v(\alpha _{v})}\end{cases}}}$

is surjective and therefore ${\displaystyle \operatorname {Cl} _{K}}$ is the continuous image of the compact set ${\displaystyle I_{K}^{1}/K^{\times }.}$ Thus ${\displaystyle \operatorname {Cl} _{K}}$ 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 ${\displaystyle 0}$ by the set of the principal divisors is a finite group.²²

Let ${\displaystyle P\supset P_{\infty }}$ be a finite set of places. Define

${\displaystyle {\begin{aligned}\Omega (P)&:=\prod _{v\in P}K_{v}^{\times }\times \prod _{v\notin P}O_{v}^{\times }=(\mathbb {A} _{K}(P))^{\times },\\E(P)&:=K^{\times }\cap \Omega (P).\end{aligned}}}$

Then ${\displaystyle E(P)}$ is a subgroup of ${\displaystyle K^{\times }}$, containing all elements ${\displaystyle \xi \in K^{\times }}$ satisfying ${\displaystyle v(\xi )=0}$ for all ${\displaystyle v\notin P.}$ Since ${\displaystyle K^{\times }}$ is discrete in ${\displaystyle I_{K}}$, ${\displaystyle E(P)}$ is a discrete subgroup of ${\displaystyle \Omega (P)}$ and, with the same argument, ${\displaystyle E(P)}$ is discrete in ${\displaystyle \Omega _{1}(P):=\Omega (P)\cap I_{K}^{1}.}$

An alternative definition is ${\displaystyle E(P)=K(P)^{\times }}$, where ${\displaystyle K(P)}$ is a subring of ${\displaystyle K}$ defined by

${\displaystyle K(P):=K\cap \left(\prod _{v\in P}K_{v}\times \prod _{v\notin P}O_{v}\right).}$

As a consequence, ${\displaystyle K(P)}$ contains all elements ${\displaystyle \xi \in K}$ which fulfil ${\displaystyle v(\xi )\geq 0}$ for all ${\displaystyle v\notin P.}$

Lemma. Let ${\displaystyle 0<c\leq C<\infty .}$ The following set is finite:

${\displaystyle \left\{\eta \in E(P):\left.{\begin{cases}|\eta _{v}|_{v}=1&\forall v\notin P,\\c\leq |\eta _{v}|_{v}\leq C&\forall v\in P.\end{cases}}\right\}\right\}.}$

Proof. Define

${\displaystyle W:=\left\{(\alpha _{v})_{v}:\left.{\begin{cases}|\alpha _{v}|_{v}=1&\forall v\notin P,\\c\leq |\alpha _{v}|_{v}\leq C&\forall v\in P.\end{cases}}\right\}\right\}.}$

${\displaystyle W}$ is compact and the set described above is the intersection of ${\displaystyle W}$ with the discrete subgroup ${\displaystyle K^{\times }}$ in ${\displaystyle I_{K}}$ and therefore finite.

Lemma. Let ${\displaystyle E}$ be set of all ${\displaystyle \xi \in K}$ such that ${\displaystyle |\xi |_{v}=1}$ for all ${\displaystyle v.}$ Then ${\displaystyle E=\mu (K)}$, the group of all roots of unity of ${\displaystyle K.}$ In particular it is finite and cyclic.

Proof. All roots of unity of ${\displaystyle K}$ have absolute value ${\displaystyle 1}$, so ${\displaystyle \mu (K)\subset E.}$ For the converse, note that the preceding lemma with ${\displaystyle c=C=1}$ and any ${\displaystyle P}$ implies ${\displaystyle E}$ is finite. Moreover ${\displaystyle E\subset E(P)}$ for each finite set of places ${\displaystyle P\supset P_{\infty }.}$ Finally, suppose there exists ${\displaystyle \xi \in E}$ which is not a root of unity of ${\displaystyle K.}$ Then ${\displaystyle \xi ^{n}\neq 1}$ for all ${\displaystyle n\in \mathbb {N} }$, contradicting the finiteness of ${\displaystyle E.}$

Unit Theorem. ${\displaystyle E(P)}$ is the direct product of ${\displaystyle E}$ and a group isomorphic to ${\displaystyle \mathbb {Z} ^{s}}$, where ${\displaystyle s=0}$ if ${\displaystyle P=\emptyset }$ and ${\displaystyle s=|P|-1}$ if ${\displaystyle P\neq \emptyset .}$²³

Dirichlet's Unit Theorem. Let ${\displaystyle K}$ be a number field. Then

${\displaystyle O^{\times }\cong \mu (K)\times \mathbb {Z} ^{r+s-1},}$

where ${\displaystyle \mu (K)}$ is the finite cyclic group of all roots of unity of ${\displaystyle K}$, ${\displaystyle r}$ is the number of real embeddings of ${\displaystyle K}$, and ${\displaystyle s}$ is the number of conjugate pairs of complex embeddings of ${\displaystyle K.}$ It stands that ${\displaystyle [K:\mathbb {Q} ]=r+2s.}$

Remark. The Unit Theorem generalises Dirichlet's Unit Theorem. To see this, let ${\displaystyle K}$ be a number field. It is already known that ${\displaystyle E=\mu (K)}$, set ${\displaystyle P=P_{\infty }}$ and note ${\displaystyle |P_{\infty }|=r+s.}$ Then there is

${\displaystyle {\begin{aligned}E\times \mathbb {Z} ^{r+s-1}=E(P_{\infty })&=K^{\times }\cap \left(\prod _{v|\infty }K_{v}^{\times }\times \prod _{v<\infty }O_{v}^{\times }\right)\\&\cong K^{\times }\cap \left(\prod _{v<\infty }O_{v}^{\times }\right)\\&\cong O^{\times }.\end{aligned}}}$

The self-duality of the adele ring extends to adelic vector spaces.

Theorem (algebraic and continuous duals of the adele ring).²⁴ Let ${\displaystyle \chi }$ be a non-trivial character of ${\displaystyle \mathbb {A} _{K}}$, which is trivial on ${\displaystyle K.}$ Let ${\displaystyle E}$ be a finite-dimensional vector space over ${\displaystyle K.}$ Let ${\displaystyle E^{\star }}$ and ${\displaystyle \mathbb {A} _{E}^{\star }}$ be the algebraic duals of ${\displaystyle E}$ and ${\displaystyle \mathbb {A} _{E}.}$ Denote the topological dual of ${\displaystyle \mathbb {A} _{E}}$ by ${\displaystyle \mathbb {A} _{E}'}$ and use ${\displaystyle \langle \cdot ,\cdot \rangle }$ and ${\displaystyle [\cdot ,\cdot ]}$ to indicate the natural bilinear pairings on ${\displaystyle \mathbb {A} _{E}\times \mathbb {A} _{E}'}$ and ${\displaystyle \mathbb {A} _{E}\times \mathbb {A} _{E}^{\star }.}$ Then the formula

${\displaystyle \langle e,e'\rangle =\chi ([e,e^{\star }])}$

for all ${\displaystyle e\in \mathbb {A} _{E}}$ determines an isomorphism ${\displaystyle e^{\star }\mapsto e'}$ of ${\displaystyle \mathbb {A} _{E}^{\star }}$ onto ${\displaystyle \mathbb {A} _{E}'}$, where ${\displaystyle e'\in \mathbb {A} _{E}'}$ and ${\displaystyle e^{\star }\in \mathbb {A} _{E}^{\star }.}$ Moreover, if ${\displaystyle e^{\star }\in \mathbb {A} _{E}^{\star }}$ fulfils ${\displaystyle \chi ([e,e^{\star }])=1}$ for all ${\displaystyle e\in E}$, then ${\displaystyle e^{\star }\in E^{\star }.}$

References

References

1. Weil 1995, Ch. IV.
2. Neukirch 1999, Ch. VI, §1.
3. Cassels & Fröhlich 1967, Ch. II.
4. Ramakrishnan & Valenza 1999, Ch. 5.
5. Tate 1967.
6. Ramakrishnan & Valenza 1999.
7. Deitmar 2010, p. 126.
8. Deitmar 2010, pp. 128–139.
9. Deitmar 2010, p. 128.
10. Neukirch 1999, Ch. VI.
11. Weil 1995, Ch. VII.
12. Cassels & Fröhlich 1967.
13. Bump 1997.
14. Deitmar 2010, Chs. 7–8.
15. Tate 1968.
16. This proof can be found in Cassels & Fröhlich 1967, p. 64.
17. The definitions are based on Weil 1967, p. 60.
18. See Weil 1967, p. 64 or Cassels & Fröhlich 1967, p. 74.
19. See Cassels & Fröhlich 1967, p. 64, Theorem, or Weil 1967, p. 64, Theorem 2.
20. A proof can be found in Cassels & Fröhlich 1967, p. 48.
21. A proof can be found in Cassels & Fröhlich 1967, p. 67.
22. For more information, see Cassels & Fröhlich 1967, p. 71.
23. A proof can be found in Weil 1967, p. 78 or in Cassels & Fröhlich 1967, p. 72.
24. A proof can be found in Weil 1967, p. 66.

Sources

Sources

• Bump, Daniel (1997), Automorphic Forms and Representations, Cambridge Studies in Advanced Mathematics, vol. 55, Cambridge University Press, ISBN 978-0-521-65818-8
• Cassels, John; Fröhlich, Albrecht (1967). Algebraic number theory: proceedings of an instructional conference, organized by the London Mathematical Society, (a NATO Advanced Study Institute). Vol. XVIII. London: Academic Press. ISBN 978-0-12-163251-9. 366 pages.
• Deitmar, Anton (2010). Automorphe Formen (in German). Vol. VIII. Berlin; Heidelberg (u.a.): Springer. ISBN 978-3-642-12389-4. 250 pages.
• Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, vol. 322, translated by Schappacher, Norbert, Springer, ISBN 978-3-540-65399-8
• Ramakrishnan, Dinakar; Valenza, Robert J. (1999), Fourier Analysis on Number Fields, Graduate Texts in Mathematics, vol. 186, Springer, ISBN 978-0-387-98436-0
• Tate, John (1967), "Fourier analysis in number fields, and Hecke's zeta-functions", in Cassels, J. W. S.; Fröhlich, Albrecht (eds.), Algebraic Number Theory, London: Academic Press, pp. 305–347
• Tate, John (1968), "Residues of differentials on curves", Annales scientifiques de l'École Normale Supérieure, 4, 1 (1): 149–159, doi:10.24033/asens.1162, Zbl 0159.22702
• Weil, André (1967). Basic number theory. Vol. XVIII. Berlin; Heidelberg; New York: Springer. ISBN 978-3-662-00048-9. 294 pages.
• Weil, André (1995), Basic Number Theory, Classics in Mathematics, Springer, ISBN 978-3-540-58655-5

External links

External links

• What problem do the adeles solve?
• Some good books on adeles