group of ideles in nLab

Contents

Contents

Idea

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

𝕀=𝔸 ×≔GL 1(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

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

𝕀 K≔GL 1(𝔸 K) \mathbb{I}_K \coloneqq GL_1(\mathbb{A}_K)

or similar. The notation J KJ_K is also common.

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

Properties

Product formula

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

|abp ℓ| p≔p −ℓ {\left \vert \frac{a}{b}p^\ell \right \vert}_p \coloneqq p^{-\ell}

for a,ba,b coprime to pp. 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 “vv” instead of pp.

This induces:

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

(e.g. Garrett 11, section 1)

Strong approximation theorem for the idele class group

(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:

𝔸 ℚ × ≃ℚ ××(ℚ ×\𝔸 ℚ ×) ≔∪n∈ℚ ×n⋅(ℚ ×\𝔸 ℚ ×). \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 FF as being the rational functions on an arithmetic curve Σ\Sigma. Under this identification the ring of adeles 𝔸 F\mathbb{A}_F of FF has the interpretation of being the ring of functions on all punctured formal disks around all points of Σ\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(𝔸 F)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 nn 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=1n = 1 then GL 1(𝔸 F)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.

• 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)