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.

The ring of adeles allows one to elegantly describe the Artin reciprocity law, which is a vast generalization of quadratic reciprocity, and other reciprocity laws over finite fields. In addition, it is a classical theorem from Weil that on an algebraic curve over a finite field can be described in terms of adeles for a reductive group .

In local class field theory, the group of units of the local field plays a central role. In global class field theory, the idele class group takes this role. The term "idele" (French: idèle) is an invention of the French mathematician Claude Chevalley (1909–1984) and stands for "ideal element" (abbreviated: id.el.). The term "adele" (adèle) stands for additive idele.

The rationals K=Q have a valuation for every prime number p, with (Kν,Oν)=(Qp,Zp), and one infinite valuation with Q=R. Thus an element of

is a real number along with a p-adic rational for each p of which all but finitely many are p-adic integers.

Secondly, take the function field K=Fq(P1)=Fq(t) of the projective line over a finite field. Its valuations correspond to points x of X=P1, i.e. maps over Spec Fq

For instance, there are q+1 points of the form SpecFqP1. In this case OνX,x is the completed stalk of the structure sheaf at x (i.e. functions on a formal neighbourhood of x) and Kν=KX,x is its fraction field. Thus

The same holds for any smooth proper curve X/Fq over a finite field, the restricted product being over all points of x∈X.

The quotient of the ideles by the subgroup K×⊆IK is called the idele class group

and its divisor group is Div(X)=AK×/OK×. Similarly, if G is a semisimple algebraic group (e.g. SLn, it also holds for GLn) then the Weil uniformisation says that[3]

There is a topology on AK for which the quotient AK/K is compact, allowing one to do harmonic analysis on it. John Tate in his thesis "Fourier analysis in number fields and Heckes Zeta functions"[4] proved results about Dirichlet L-functions using Fourier analysis on the adele ring and the idele group. Therefore, the adele ring and the idele group have been applied to study the Riemann zeta function and more general zeta functions and the L-functions.

If X is a smooth proper curve over the complex numbers, one can define the adeles of its function field C(X) exactly as the finite fields case. John Tate proved[5] that Serre duality on X

can be deduced by working with this adele ring AC(X). Here L is a line bundle on X.

It is equipped with the restricted product topology, the topology generated by restricted open rectangles, which have the following form:

Remark. Global function fields do not have any infinite places and therefore the finite adele ring equals the adele ring.

They are continuous maps on the adele ring and they fulfil the usual equations:

Remark. The fourth statement is a special case of the strong approximation theorem.

We equip the idele group with the topology defined in the Lemma making it a topological group.

In the previous section we used the fact that the class number of a number field is finite. Here we would like to prove this statement:

The theory of automorphic forms is a generalization of Tate's thesis by replacing the idele group with analogous higher dimensional groups. To see this note:

Based on these identification a natural generalization would be to replace the idele group and the 1-idele with:

The idele class group is a key object of class field theory, which describes abelian extensions of the field. The product of the local reciprocity maps in local class field theory gives a homomorphism of the idele group to the Galois group of the maximal abelian extension of the global field. The Artin reciprocity law, which is a high level generalisation of the Gauss quadratic reciprocity law, states that the product vanishes on the multiplicative group of the number field. Thus, we obtain the global reciprocity map of the idele class group to the abelian part of the absolute Galois group of the field.

The self-duality of the adele ring of the function field of a curve over a finite field easily implies the Riemann–Roch theorem and the duality theory for the curve.