nLab ring of adeles

Idea

The ring of adeles $\mathbb{A}_k$ of any global field $k$ – in particular of the rational numbers $\mathbb{Q}$ – is the restricted product of all formal completions $k_v$ of $k$ at all its places $v$, where the restriction is such that only a finite number of components have norm greater than 1. (This has a useful geometric interpretation and motivation by the function field analogy, more on which below).

In particular the ring of adeles of the rational numbers is equivalently the rationalization of the product of all p-adic integers (including the “prime at infinity”).

In classical algebraic number theory one embeds a number field into the cartesian product of its completions at its archimedean absolute values. This embedding is very useful in the proofs of several fundamental theorems. For example, the algebraic integers in the number field embed discretely and co-compactly into this cartesian product, i.e., as a lattice, and this opens the way for example to the concrete realization of the group of units (modulo torsion) as a lattice, and also to the technique of Fourier analysis where Poisson summation applied to the lattice has classical implications for theta functions and zeta functions.

It was noticed by Claude Chevalley and André Weil that the situation is made even better if the number field itself is embedded in the cartesian product of its formal completions at all of its absolute values. With a few additional restrictions, these objects are known as the adeles, and the units of this ring are called the ideles. Under these restrictions, the given number field embeds discretely and cocompactly into the adeles, i.e., behaves as a lattice where it is again possible to apply Poisson summation.

When considering the adeles and ideles, it is their topology as much as their algebraic structure that is of interest. Many important results in number theory translate into simple statements about the topologies of the adeles and ideles. For example, the finiteness of the ideal class group and the Dirichlet unit theorem are equivalent to a certain quotient of the ideles being compact and discrete.

(Weston, p. 1)

Definition

The concept of a ring of adeles (older synonym: ring of valuation vectors) makes sense for any global field, hence for any finite-dimensional field extension of either the rational numbers or of a function field over a finite field. The ring of adeles of the rational numbers themselves is the classical case which we discuss first in

• For the rational numbers

Then we consider the definition more generally

• For a number field

and finally in full generality

• For a global field.

For the rational numbers

We start off very simply with the algebraic description of the adele ring over the rational numbers.

Definition

The ring of integral adeles $\mathbb{A}_{\mathbb{Z}}$ is the product of the profinite completion of the integers $\widehat{\mathbb{Z}}$, with the real numbers

$$\mathbb{A}_{\mathbb{Z}} \coloneqq \mathbb{R} \times \widehat{\mathbb{Z}} \,.$$

The ring of adeles $\mathbb{A}_{\mathbb{Q}}$ (or just $\mathbb{A}$, for short) itself is the rationalization of the ring of integral adeles, hence its tensor product with the rational numbers

$$\mathbb{A}_\mathbb{Q} \coloneqq \mathbb{Q} \otimes_\mathbb{Z} \mathbb{A}_\mathbb{Z} \,.$$

This definition has various equivalent reformulations which are often useful.

Remark

By this proposition we have that the profinite completion of the integers is equivalently the product of all p-adic integers as $p$ ranges over all prime numbers

$$\hat \mathbb{Z} \simeq \underset{p\;prime}{\prod} \mathbb{Z}_p$$

Using this in def. says that the ring of integral adeles is the product

$$\mathbb{A}_{\mathbb{Z}} \simeq \mathbb{R} \times \underset{p\;prime}{\prod} \mathbb{Z}_p \,.$$

From this one obtains the following equivalent characterization:

Proposition

The ring of adeles $\mathbb{A}$, def. , is equivalently the restricted product $\prod^\prime$ of the p-adic rational numbers, the restriction being along the inclusion $\mathbb{Z}_p \to \mathbb{Q}_p$:

$$\mathbb{A}_{\mathbb{Q}} = \mathbb{R} \times \underset{p \; prime}{\prod^\prime} \mathbb{Q}_p$$

Proof

By remark the tensor product to be computed is equivalently

$$\mathbb{A}_{\mathbb{Q}} \simeq \mathbb{Q}\otimes_{\mathbb{Z}} \left( \mathbb{R} \times \underset{p \; prime}{\prod} \mathbb{Z}_p \right) \,.$$

Now notice that a natural number $n$ is a unit in $\mathbb{Z}_p$ if $p$ is not a prime factor of $n$. Therefore for $(a_p) \in \underset{p}{\prod} \mathbb{Z}_p$ and $\frac{c}{d} \in \mathbb{Q}$, then for each of the finite number of prime factors $p$ of $d$ the tensor product element $\frac{c}{d} \otimes_{\mathbb{Z}} a_p \in \mathbb{Q}_p$ contains a non-vanishing negative power of $p$ and is hence not in $\mathbb{Z}_p$, whereas for all $p$ that do not appear as prime factors in $d$ it is.

Finally notice:

Remark

The prime numbers correspond to the non-archimedean places of $\mathbb{Z}$, and under this identification there is one more real place at infinity, “$p = \infty$”, the completion of $\mathbb{Q}$ at which is the real numbers $\mathbb{R}$, which one may therefore write $\mathbb{R} = \mathbb{Q}_\infty$. Using this the characterization of the ring of adeles from prop. is equivalently the restricted product over all real places of the formal completion of $\mathbb{Q}$ at this place

$$\mathbb{A}_{\mathbb{Q}} \simeq \underset{p \in Places(\mathbb{Z})}{\prod^{\prime}} \mathbb{Q}_p \,.$$

Considering this restricted product not just in bare commutative rings but in topological rings yields the right structure of a topological ring on $\mathbb{A}_{\mathbb{Q}}$. This is the content of the following proposition.

Proposition

$\mathbb{A}_\mathbb{Q}$ is a locally compact Hausdorff commutative ring. In particular, it is complete with respect to its uniform space structure.

Proof

The restricted product is a filtered colimit of a system of open inclusions between locally compact Hausdorff rings, and is therefore itself locally compact Hausdorff. If $x_\alpha$ is a Cauchy net, then for all sufficiently large $\alpha, \beta$ the differences $x_\alpha - x_\beta$ lie in a compact neighborhood of the identity. Holding $\beta$ fixed, the limit $\lim_\alpha x_\alpha - x_\beta$ exists by compactness; if $x$ is this limit, then $x + x_\beta$ is the limit of the Cauchy net.

If one omits the factor of $\mathbb{R} = \mathbb{Q}_\infty$, then one speaks of the ring of finite adeles.

$$\mathbb{A}_{\mathbb{Q}}^f \coloneqq \underset{p \; prime}{\prod^\prime} \mathbb{Q}_p \,.$$

All of this generalizes to any number field $k$.

For a number field

Let

• $k$ be a number field;

• $\mathcal{O} = \mathcal{O}_k$ the ring of algebraic integers in $k$:

• $P$ be its set of places (equivalence classes of absolute values on $\mathcal{O}$);

• $S\subset P$ be the set of archimedean places;

• $k_v$ the formal completion of $k$ at $v\in P$.

Notice that

• for $v \in S$ then $k_v$ is isomorphic to one of the local fields $\mathbb{R}$ or $\mathbb{C}$;

• for $v \notin S$ then $k_v$ is a local field with an open compact subring $\mathcal{O}_v$ consisting of elements of norm $1$ or less.

Definition

For $k$ a number field, the ring of integral adeles $\mathbb{A}_{\mathcal{O}}$ is the product of the profinite completion $\widehat{\mathcal{O}}$ with all the archimedean completions,

$$\mathbb{A}_\mathcal{O} \coloneqq \widehat{\mathcal{O}} \times \prod_{v \in S} k_v \,,$$

and the ring of adeles over $k$ is the tensor product $\mathbb{A}_k \coloneqq k \otimes_\mathcal{O} \mathbb{A}_\mathcal{O}$.

It may be shown that

$$\widehat{\mathcal{O}} \cong \prod_{v \notin S} \mathcal{O}_v$$

where each non-archimedean place $v$ may be identified with a prime ideal $p$. Now, if we view $k$ as the localization of $\mathcal{O}$ obtained by inverting all nonzero elements $x \in \mathcal{O}$, then $k$ may be written as a filtered colimit of a system of inclusions of localizations:

$$k \cong colim_x \; \mathcal{O}[x^{-1}].$$

Componentwise, we may calculate

$$\array{ \mathcal{O}[x^{-1}] \otimes_\mathcal{O} \mathcal{O}_p & = & k_p & \text{if}\; x \in p \\ & = & \mathcal{O}_p & \text{if}\; x \notin p }$$

and so

$$\array{ \mathcal{O}[x^{-1}] \otimes_\mathcal{O} \widehat{\mathcal{O}} & \cong & \mathcal{O}[x^{-1}] \otimes_\mathcal{O} \prod_{p \in P \backslash\; S} \mathcal{O}_p \\ & \cong & (\prod_{x \in p} k_p) \times (\prod_{x \notin p} \mathcal{O}_p) }$$

Putting these facts together,

$$k \otimes_\mathcal{O} \widehat{\mathcal{O}} \cong colim_x\; (\prod_{x \in p} k_p) \times (\prod_{x \notin p} \mathcal{O}_p)$$

whence

$$\begin{array}{lll} k \otimes_\mathcal{O} \mathbb{A}_\mathcal{O} & \cong & k \otimes_\mathcal{O} \left(\prod_{v \in S} k_v \times \widehat{\mathcal{O}}\right) \\ & \cong & colim_x \; \prod_{v \in S} k_v \times (\prod_{p: x \in p} k_p) \times (\prod_{p: x \notin p} \mathcal{O}_p) \end{array}$$

where each component of the filtered diagram is locally compact (a product of finitely many locally compact and infinitely many compact spaces) in the product topology. Taking the filtered colimit in $Top$ over the resulting diagram of open inclusions, the result is again a locally compact ring. In this way the ring of adeles $\mathbb{A}_k$ is topologized.

Remark

The topology on the adele ring $\mathbb{A}_k$ is strictly finer than the subspace topology inherited from its natural inclusion into $\prod_{v \in P} k_v$ with the product topology. For example, $(\prod_{v \in S} k_v) \times \prod_{p \in P \backslash\; S} \mathcal{O}_p$ is open in the ring of adeles, but not in $\prod_{v \in P} k_v$.

Definition

The group of units of the ring of adeles $\mathbb{A}_k$ is called the group of ideles, denoted $\mathbb{I}_k$.

Under the subspace topology inherited from $\mathbb{A}_k$, there is no reason for inversion $(-)^{-1}: \mathbb{I}_k \to \mathbb{I}_k$ to be continuous (and in fact it isn’t!), so $\mathbb{I}_k$ isn’t a topological group when topologized this way. However, we can endow $\mathbb{I}_k$ with the subspace topology given by the embedding $\mathbb{I}_k \to \mathbb{A}_k \times \mathbb{A}_k: x \mapsto (x, x^{-1})$; topologized this way, we get a locally compact topological group. This is the topology on the ideles.

Alternatively, for each finite $T \subset P$ containing the set of archimedean places $S$, we have a locally compact group

$$\mathbb{I}_{k, T} = (\prod_{v \in T} k_v^\times) \times (\prod_{v \notin T} \mathcal{O}_v^\times)$$

(noting that each unit group $\mathcal{O}_v^\times$ is compact), and the idele group can be described as the colimit over a filtered system of open inclusions

$$\mathbb{I}_k = colim_T \; \mathbb{I}_{k, T}$$

and indeed the idele topology coincides with the filtered colimit topology.

For a global field

Fully generally, let $k$ be a global field. Write $P$ for its set of places and $k_v$ for its formal completion at $v \in P$.

Definition

The ring of adeles of a global field $k$ is the restricted product

$$\mathbb{A}_k \coloneqq \underset{v \in P}{\prod} k_v$$

where the restriction is to elements $(x_v)_{v\in P}$ of the actual product whose components have norm at most unity – ${\vert x_v\vert} \leq 1$, except for at most a finite number of $v$.

This is topologized in the same way as discussed above.

Reviews includes (Mathew 10).

In the function field case, where $k$ is a finite extension of $\mathbb{F}_p(T)$, the analogous ring of integers $\mathcal{O}(k)$ is the integral closure in $k$ of the subring $\mathbb{F}_p[T] \hookrightarrow \mathbb{F}_p(T)$. And analogously, the ring of integer adeles $\mathbb{A}_{\mathcal{O}(k)}$ may be defined to be the product of all the completions of $\mathcal{O}(k)$ over all the places of $k$. This is a compact ring. The restricted direct product above map may then, in parallel with the number field case described above, be described as a tensor product

$$\mathbb{A}_k = k \otimes_{\mathcal{O}(k)} \mathbb{A}_{\mathcal{O}(k)}$$

where the right side is again interpreted as a colimit in the category of topological rings of a diagram consisting of compact topological rings and open inclusions between them.

Properties

Basic results

Proposition

Under the natural inclusion $i: k \to \mathbb{A}_k$, the subspace topology on $k$ is discrete, and the quotient topology on $\mathbb{A}_k/k$ is compact.

As an additive topological group, there is a natural pairing on $\mathbb{A}_k$:

$$\langle -, - \rangle: \mathbb{A}_k \times \mathbb{A}_k \to \mathbb{R}/\mathbb{Z} \cong S^1.$$

If $x = (x_v)_{v \in P}$ and $y = (y_v)_{v \in P}$, then

$$\langle x, y \rangle \coloneqq \prod_v \langle x_v, y_v \rangle_v$$

where each local pairing $\langle -, - \rangle_v$ is defined to be a composite of the form

$$k_v \times k_v \stackrel{mult}{\to} k_v \stackrel{Tr}{\to} \mathbf{Q}_v \stackrel{\chi_v}{\to} \mathbb{R}/\mathbb{Z} \cong S^1,$$

noting that a place $v$ of $k$ restricts to a place on $\mathbb{Q}$.

Remarks

• The trace map $Tr$ on the finite algebraic extension $k_v/\mathbf{Q}_v$ is of course defined by $Tr(x) = \sum_\sigma \sigma(x)$ where $\sigma$ ranges over all embeddings of $k_v$ into the algebraic closure of $\mathbb{Q}_v$.

• When $v$ is the archimedean place on $\mathbb{Q}$, we will take the map $\chi_v: \mathbb{R} \to \mathbb{R}/\mathbb{Z}$ to be not the quotient map, but its additive inverse. For non-archimedean places $v = p$, the character $\chi_p$ is the composite $\mathbf{Q}_p \to \mathbb{Z}(p^\infty) \hookrightarrow \mathbb{R}/\mathbb{Z}$ as defined here.

• For each $x, y \in \mathbb{A}_k$, observe that $\langle x_v, y_v \rangle_v = 1 \in S^1$ for all but finitely many places $v$, since $x_v, y_v \in \mathcal{O}_v$ for all but finitely many places. Hence $\langle x, y\rangle$ is well-defined.

Proposition

The additive group $\mathbb{A}_k$ is Pontrjagin self-dual in the sense that the map $\phi: \mathbb{A}_k \to \mathbb{A}_k^\wedge$ induced from the pairing $\langle-, -\rangle$ is an isomorphism onto the character group.

Moreover, define $\pi: \mathbb{A}_k \to k^\wedge$ to be the composite

$$\mathbb{A}_k \stackrel{\phi}{\to} \mathbb{A}_k^\wedge \stackrel{i^\wedge}{\to} k^\wedge.$$

Proposition

The map $\pi \circ i: k \to k^\wedge$ vanishes. The map $\mathbb{A}_k/k \to k^\wedge$ induced by $\pi: \mathbb{A}_k \to k^\wedge$ is an isomorphism of topological groups, so that $\mathbb{A}_k/k$ is Pontrjagin dual to $k$.

Function field analogy

Via the function field analogy one may understand any number field or function field $F$ as being the rational functions on an arithmetic curve $\Sigma$. Under this identification the ring of adeles $\mathbb{A}_F$ of $F$ has the interpretation of being the ring of functions on all punctured formal disks around all points of $\Sigma$, such that only finitely many of them do not extend to the given point. (Frenkel 05, section 3.2). This is most manifest in terms of def. above.

This means for instance that the general linear group $GL_n(\mathbb{A}_F)$ with coefficients in the ring of adeles has the interpretation as being the Cech cocycles for algebraic vector bundles of rank $n$ on an algebraic curve with respect to any cover of that curve by the complement of a finite number of points together with the formal disks around these points. (Notice that for $n = 1$ then $GL_1(\mathbb{A}_F)$ is the group of ideles). This is part of a standard construction of the moduli stack of bundles on algebraic curves, see at Moduli space of bundles and the Langlands correspondence.

function field analogy

	number fields (“function fields of curves over F1”)	function fields of curves over finite fields $\mathbb{F}_q$ (arithmetic curves)	Riemann surfaces/complex curves
affine and projective line
	$\mathbb{Z}$ (integers)	$\mathbb{F}_q[z]$ (polynomials, polynomial algebra on affine line $\mathbb{A}^1_{\mathbb{F}_q}$)	$\mathcal{O}_{\mathbb{C}}$ (holomorphic functions on complex plane)
	$\mathbb{Q}$ (rational numbers)	$\mathbb{F}_q(z)$ (rational fractions/rational function on affine line $\mathbb{A}^1_{\mathbb{F}_q}$)	meromorphic functions on complex plane
	$p$ (prime number/non-archimedean place)	$x \in \mathbb{F}_p$, where $z - x \in \mathbb{F}_q[z]$ is the irreducible monic polynomial of degree one	$x \in \mathbb{C}$, where $z - x \in \mathcal{O}_{\mathbb{C}}$ is the function which subtracts the complex number $x$ from the variable $z$
	$\infty$ (place at infinity)		$\infty$
	$Spec(\mathbb{Z})$ (Spec(Z))	$\mathbb{A}^1_{\mathbb{F}_q}$ (affine line)	complex plane
	$Spec(\mathbb{Z}) \cup place_{\infty}$	$\mathbb{P}_{\mathbb{F}_q}$ (projective line)	Riemann sphere
	$\partial_p \coloneqq \frac{(-)^p - (-)}{p}$ (Fermat quotient)	$\frac{\partial}{\partial z}$ (coordinate derivation)	“
	genus of the rational numbers = 0		genus of the Riemann sphere = 0
formal neighbourhoods
	$\mathbb{Z}/(p^n \mathbb{Z})$ (prime power local ring)	$\mathbb{F}_q [z]/((z-x)^n \mathbb{F}_q [z])$ ($n$-th order univariate local Artinian $\mathbb{F}_q$-algebra)	$\mathbb{C}[z]/((z-x)^n \mathbb{C}[z])$ ($n$-th order univariate Weil $\mathbb{C}$-algebra)
	$\mathbb{Z}_p$ (p-adic integers)	$\mathbb{F}_q[ [ z -x ] ]$ (power series around $x$)	$\mathbb{C}[ [z-x] ]$ (holomorphic functions on formal disk around $x$)
	$Spf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X$ (“$p$-arithmetic jet space” of $X$ at $p$)		formal disks in $X$
	$\mathbb{Q}_p$ (p-adic numbers)	$\mathbb{F}_q((z-x))$ (Laurent series around $x$)	$\mathbb{C}((z-x))$ (holomorphic functions on punctured formal disk around $x$)
	$\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p$ (ring of adeles)	$\mathbb{A}_{\mathbb{F}_q((t))}$ ( adeles of function field )	$\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((z-x))$ (restricted product of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks)
	$\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}})$ (group of ideles)	$\mathbb{I}_{\mathbb{F}_q((t))}$ ( ideles of function field )	$\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x)))$
theta functions
	Jacobi theta function
zeta functions
	Riemann zeta function	Goss zeta function
branched covering curves
	$K$ a number field ($\mathbb{Q} \hookrightarrow K$ a possibly ramified finite dimensional field extension)	$K$ a function field of an algebraic curve $\Sigma$ over $\mathbb{F}_p$	$K_\Sigma$ (sheaf of rational functions on complex curve $\Sigma$)
	$\mathcal{O}_K$ (ring of integers)		$\mathcal{O}_{\Sigma}$ (structure sheaf)
	$Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z})$ (spectrum with archimedean places)	$\Sigma$ (arithmetic curve)	$\Sigma \to \mathbb{C}P^1$ (complex curve being branched cover of Riemann sphere)
	$\frac{(-)^p - \Phi(-)}{p}$ (lift of Frobenius morphism/Lambda-ring structure)	$\frac{\partial}{\partial z}$	“
	genus of a number field	genus of an algebraic curve	genus of a surface
formal neighbourhoods
	$v$ prime ideal in ring of integers $\mathcal{O}_K$	$x \in \Sigma$	$x \in \Sigma$
	$K_v$ (formal completion at $v$)		$\mathbb{C}((z_x))$ (function algebra on punctured formal disk around $x$)
	$\mathcal{O}_{K_v}$ (ring of integers of formal completion)		$\mathbb{C}[ [ z_x ] ]$ (function algebra on formal disk around $x$)
	$\mathbb{A}_K$ (ring of adeles)		$\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x))$ (restricted product of function rings on all punctured formal disks around all points in $\Sigma$)
	$\mathcal{O}$		$\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ]$ (function ring on all formal disks around all points in $\Sigma$)
	$\mathbb{I}_K = GL_1(\mathbb{A}_K)$ (group of ideles)		$\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))$
Galois theory
	Galois group	“	$\pi_1(\Sigma)$ fundamental group
	Galois representation	“	flat connection (“local system”) on $\Sigma$
class field theory
	class field theory	“	geometric class field theory
	Hilbert reciprocity law	Artin reciprocity law	Weil reciprocity law
	$GL_1(K)\backslash GL_1(\mathbb{A}_K)$ (idele class group)	“
	$GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})$	“	$Bun_{GL_1}(\Sigma)$ (moduli stack of line bundles, by Weil uniformization theorem)
non-abelian class field theory and automorphy
	number field Langlands correspondence	function field Langlands correspondence	geometric Langlands correspondence
	$GL_n(K) \backslash GL_n(\mathbb{A}_K)//GL_n(\mathcal{O})$ (constant sheaves on this stack form unramified automorphic representations)	“	$Bun_{GL_n(\mathbb{C})}(\Sigma)$ (moduli stack of bundles on the curve $\Sigma$, by Weil uniformization theorem)
	Tamagawa-Weil for number fields	Tamagawa-Weil for function fields
theta functions
	Hecke theta function		functional determinant line bundle of Dirac operator/chiral Laplace operator on $\Sigma$
zeta functions
	Dedekind zeta function	Weil zeta function	zeta function of a Riemann surface/of the Laplace operator on $\Sigma$
higher dimensional spaces
zeta functions	Hasse-Weil zeta function

Related concepts

• adelic string theory

• adelic integration

References

• wikipedia: adele ring, adelic algebraic group

• D. Goldfeld, J. Hundley, chapter 1 of Automorphic Representations and L-functions for the General Linear Group, vol. 1, Cambridge University Press, 2011 (pdf)

• Akhil Mathew, The adele ring, 2010

• Adeles pdf

• Pete Clark, Adeles and Ideles (pdf)

• Erwin Dassen , Adeles & Ideles (pdf)

• Tom Weston, The idelic approach to number theory (pdf)

• MO/96137/categorical-description-of-the-restricted-product-adeles

Discussion in the context of the function field analogy and the geometric Langlands correspondence is in

• Edward Frenkel, Lectures on the Langlands Program and Conformal Field Theory, in Frontiers in number theory, physics, and geometry II, Springer Berlin Heidelberg, 2007. 387-533. (arXiv:hep-th/0512172)