Adele ring

In mathematics, the adele ring of a global field (also adelic ring, ring of adeles or ring of adèles^[1]) is a central object of class field theory, a branch of algebraic number theory. It is the restricted product of all the completions of the global field and is an example of a self-dual topological ring.

This section is about the concept in mathematics. For the singer, see Adele.

An adele derives from a particular kind of idele. "Idele" derives from the French "idèle" and was coined by the French mathematician Claude Chevalley. The word stands for 'ideal element' (abbreviated: id.el.). Adele (French: "adèle") stands for 'additive idele' (that is, additive ideal element).

The ring of adeles allows one to describe the Artin reciprocity law, which is a generalisation of quadratic reciprocity, and other reciprocity laws over finite fields. In addition, it is a classical theorem from Weil that $G$ -bundles on an algebraic curve over a finite field can be described in terms of adeles for a reductive group $G$ . Adeles are also connected with the adelic algebraic groups and adelic curves.

The study of geometry of numbers over the ring of adeles of a number field is called adelic geometry.

For each element of $K$ the valuations are zero for almost all places, i.e., for all places except a finite number. So, the global field can be embedded in the restricted product.

The restricted product is a , while the Cartesian product is not. Therefore, there cannot be any application of harmonic analysis to the Cartesian product. This is because local compactness ensures the existence (and uniqueness) of Haar measure, a crucial tool in analysis on groups in general.

locally compact space

Examples[edit]

Ring of adeles for the rational numbers[edit]

The rationals $K={\mathbf {Q}}$ have a valuation for every prime number $p$ , with $(K_{\nu },{\mathcal {O}}_{\nu })=(\mathbf {Q} _{p},\mathbf {Z} _{p})$ , and one infinite valuation ∞ with $\mathbf {Q} _{\infty }=\mathbf {R}$ . Thus an element of

Applications[edit]

Stating Artin reciprocity[edit]

The Artin reciprocity law says that for a global field $K$ ,

Notation and basic definitions[edit]

Global fields[edit]

Throughout this article, $K$ is a global field, meaning it is either a number field (a finite extension of $\mathbb {Q}$ ) or a global function field (a finite extension of $\mathbb {F} _{p^{r}}(t)$ for $p$ prime and $r\in \mathbb {N}$ ). By definition a finite extension of a global field is itself a global field.

Valuations[edit]

For a valuation $v$ of $K$ it can be written $K_{v}$ for the completion of $K$ with respect to $v.$ If $v$ is discrete it can be written $O_{v}$ for the valuation ring of $K_{v}$ and ${\mathfrak {m}}_{v}$ for the maximal ideal of $O_{v}.$ If this is a principal ideal denoting the uniformising element by $\pi _{v}.$ A non-Archimedean valuation is written as $v<\infty$ or $v\nmid \infty$ and an Archimedean valuation as $v|\infty .$ Then assume all valuations to be non-trivial.

There is a one-to-one identification of valuations and absolute values. Fix a constant $C>1,$ the valuation $v$ is assigned the absolute value $|\cdot |_{v},$ defined as:

Applications[edit]

Finiteness of the class number of a number field[edit]

In the previous section the fact that the class number of a number field is finite had been used. Here this statement can be proved:

; 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.

Cassels, John

(2007). Algebraische Zahlentheorie, unveränd. nachdruck der 1. aufl. edn (in German). Vol. XIII. Berlin: Springer. ISBN 9783540375470. 595 pages.

Neukirch, Jürgen

(1967). Basic number theory. Vol. XVIII. Berlin; Heidelberg; New York: Springer. ISBN 978-3-662-00048-9. 294 pages.

Weil, André

Deitmar, Anton (2010). Automorphe Formen (in German). Vol. VIII. Berlin; Heidelberg (u.a.): Springer. 978-3-642-12389-4. 250 pages.

ISBN

(1994). Algebraic number theory, Graduate Texts in Mathematics 110 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-94225-4.

Adele ring

locally compact space

Examples[edit]

Ring of adeles for the rational numbers[edit]

Applications[edit]

Stating Artin reciprocity[edit]

Notation and basic definitions[edit]

Global fields[edit]

Valuations[edit]

Applications[edit]

Finiteness of the class number of a number field[edit]

Cassels, John

Neukirch, Jürgen

Weil, André

ISBN

Lang, Serge

What problem do the adeles solve?

Some good books on adeles