nLab group of ideles

Idea

The group of ideles $\mathbb{I}$ is the group of units in the ring of adeles $\mathbb{A}$:

$$\mathbb{I} = \mathbb{A}^\times \coloneqq GL_1(\mathbb{A}) \,.$$

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. However, it was noticed by Claude Chevalley and André Weil that the situation was improved somewhat if the number field is embedded in a certain restricted product of its formal completions at all of its absolute values. These objects are known as the adeles, and the units of this ring are called the ideles.

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

Definition

The group of units of the ring of adeles $\mathbb{A}_{\mathbb{Q}}$ is called the group of ideles

$$\mathbb{I}_{\mathbb{Q}} \coloneqq GL_1(\mathbb{A}_{\mathbb{Q}}) = \mathbb{A}_{\mathbb{Q}}^\times \,.$$

It is a topological group via identification with the set $\{(x, x^{-1}) \in \mathbb{A}_\mathbb{Q}^2: \; x \in \mathbb{I}_\mathbb{Q}\}$, seen as a subspace of $\mathbb{A}_\mathbb{Q}^2$.

Remark

The topology on $\mathbb{I}_\mathbb{Q}$ is strictly finer than the subspace topology inherited from $\mathbb{A}_\mathbb{Q}$. For example, the set $\mathbb{R}^\times \times \prod_p \mathbb{Z}_p^\times$ is a neighborhood of $1$ in $\mathbb{I}_\mathbb{Q}$, but not in the subspace topology. Cf. the discussion here. Note: multiplicative inversion is not continuous in the subspace topology.

The same definition holds for the ring of adeles of any other global field $K$, here one writes

$$\mathbb{I}_K \coloneqq GL_1(\mathbb{A}_K)$$

or similar. The notation $J_K$ is also common.

Definition

The quotient

$$K^\times \backslash \mathbb{I}_K = GL_1(K)\backslash GL_1(\mathbb{A}_K)$$

is called the idele class group of $K$.

Remark

The idele class group, def. , appears prominently in the description of the moduli space of line bundles over the arithmetic curve on which $K$ is the rational functions. From there it appears in the abelian Langlands correspondence and in the abelian case of Tamagawa measures.

The idele class group is a key object in class field theory.

Properties

Product formula

Recall the p-adic norm ${\vert -\vert}_p$ on $\mathbb{Q}$ for $p$ a prime number, given by

$${\left \vert \frac{a}{b}p^\ell \right \vert}_p \coloneqq p^{-\ell}$$

for $a,b$ coprime to $p$. The usual absolute value norm one writes

$${\vert -\vert}_\infty$$

and associates with the “prime at infinity”. When an index runs over the set of all primes (“finite primes”) union with the “prime at infinity” one usually writes it “$v$” instead of $p$.

This induces:

Definition

The idele norm

$${\vert -\vert} \colon \mathbb{I}_{\mathbb{Q}} \longrightarrow \mathbb{C}^\times$$

is the function given by

$${\vert \alpha\vert} \coloneqq \underset{v}{\prod} {\vert \alpha_v\vert}_v \,.$$

Notice that by construction there is a diagonal map $\mathbb{Q}^\times \to \mathbb{I}_{\mathbb{Q}}$.

Proposition

(product formula)

The idele norm, def. , is trivial on the diagonal of $\mathbb{Q}^\times$ inside the ideles, in that

$$(\alpha \in \mathbb{Q}^\times \to \mathbb{I}_{\mathbb{Q}}) \;\Rightarrow\; {\vert \alpha\vert} \coloneqq \underset{v}{\prod} {\vert \alpha_v\vert}_v = 1 \,.$$

Remark

The product formula, prop. , says that the idele norm descends to the idele class group, def. .

(e.g. Garrett 11, section 1)

Strong approximation theorem for the idele class group

Proposition

(strong approximation form ideles)

The idele class group, def. , may be expressed as

$$\mathbb{Q}^\times \backslash \mathbb{A}_{\mathbb{Q}}^\times \simeq (0,\infty) \times \underset{p}{\prod} \mathbb{Z}_p^\times \,.$$

(e.g. Goldfeld-Hundley 11, prop. 1.4.5 and below (2.2.7))

This implies that the ring of adeles may be decomposed into a rational and an idele class factor as:

$$\begin{aligned} \mathbb{A}_{\mathbb{Q}}^\times & \simeq \mathbb{Q}^\times \times (\mathbb{Q}^\times \backslash \mathbb{A}_{\mathbb{Q}}^\times) \\ & \coloneqq \underset{n \in \mathbb{Q}^\times}{\cup} n \cdot (\mathbb{Q}^\times \backslash \mathbb{A}_{\mathbb{Q}}^\times) \end{aligned} \,.$$

(e.g. Goldfeld-Hundley 11, prop. 1.4.6 and below (2.2.7))

This decomposition is crucial in the discussion of the Riemann zeta function (see there) as an adelic integral.

Automorphic forms and Relation to Dirichlet characters

The automorphic forms of the idele group are essentially Dirichlet characters in disguise (Goldfeld-Hundley 11, below def. 2.1.4)

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 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. Here 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

• differential cohesion and idelic structure

References

Basics are recalled in

• Adeles pdf

• Pete Clark, Adeles and Ideles (pdf)

• Erwin Dassen , Adeles & Ideles (pdf)

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

• Dorian Goldfeld, Joseph Hundley, chapter 2 of Automorphic representations and L-functions for the general linear group, Cambridge Studies in Advanced Mathematics 129, 2011 (pdf)

• Paul Garrett, Iwasawa-Tate on ζ-functions and L-functions, 2011 (pdf

Discussion in the context of the geometric Langlands correspondence is in

• Edward Frenkel, section 3.2 of 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)