symmetric monoidal (∞,1)-category of spectra
The group of ideles $\mathbb{I}$ is the group of units in the ring of adeles $\mathbb{A}$:
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 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.
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.
The group of units of the ring of adeles $\mathbb{A}_{\mathbb{Q}}$ is called the group of ideles
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 $\mathbb{A}_\mathbb{Q}^2$.
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
or similar. The notation $J_K$ is also common.
The quotient
is called the idele class group of $K$.
The idele class group, def. 2, 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.
Recall the p-adic norm ${\vert -\vert}_p$ on $\mathbb{Q}$ for $p$ a prime number, given by
for $a,b$ coprime to $p$. The usual absolute value norm one writes
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:
The idele norm
is the function given by
Notice that by construction there is a diagonal map $\mathbb{Q}^\times \to \mathbb{I}_{\mathbb{Q}}$.
(product formula)
The idele norm, def. 3, is trivial on the diagonal of $\mathbb{Q}^\times$ inside the ideles, in that
The product formula, prop. 1, says that the idele norm descends to the idele class group, def. 2.
(e.g. Garrett 11, section 1)
(strong approximation form ideles)
The idele class group, def. 2, may be expressed as
(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:
(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.
The automorphic forms of the idele group are essentially Dirichlet characters in disguise (Goldfeld-Hundley 11, below def. 2.1.4)
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.
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, function 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 functions) | meromorphic functions on complex plane | |
$p$ (prime number/non-archimedean place) | $x \in \mathbb{F}_p$ | $x \in \mathbb{C}$ | |
$\infty$ (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 | |||
$\mathbb{Z}_p$ (p-adic integers) | $\mathbb{F}_q[ [ t -x ] ]$ (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 |
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