In algebraic number theory and topological algebra, the adele ring (other names are the adelic ring, the ring of adeles) is a self-dual topological ring built on the field of rational numbers (or, more generally, any algebraic number field). It involves in a symmetric way all the completions of the field.
The adele ring and its relation to the number field are among the most fundamental objects in number theory. The quotient of its multiplicative group by the multiplicative group of the algebraic number field is the central object in class field theory. It is a central principle of Diophantine geometry to study solutions of polynomials equations in number fields by looking at their solutions in the larger complete adele ring, where it is generally easier to detect solutions, and then deciding which of them come from the number field.
The word "adele" is short for "additive idele" and it was invented by André Weil. The previous name was the valuation vectors. The ring of adeles was historically preceded by the ring of repartitions, a construction which avoids completions, and is today sometimes referred to as pre-adele.
The ring of integral adeles AZ is the product
The ring of (rational) adeles AQ is the tensor product
(topologized so that AZ is an open subring).
More generally the ring of adeles AF of any algebraic number field F is the tensor product
The ring of (rational) adeles can also be defined as the restricted product
of all the p-adic completions Qp and the real numbers (or in other words as the restricted product of all completions of the rationals). In this case the restricted product means that for an adele (a∞, a2, a3, a5, …) all but a finite number of the ap are p-adic integers.
The adeles of a function field over a finite field can be defined in a similar way, as the restricted product of all completions.
The addivite group of the adele ring is a locally compact complete group with respect to its most natural topology. This group is self dual in the sense that it is topologically isomorphic to its group of characters. The adelic ring contains the number or function field as a discrete co-compact subgroup. Similarly, the multiplicative group of adeles, called the group of ideles, is a locally compact group with respect to its topology defined below.
The group of invertible elements of the adele ring is the idele group. It is not given the subset topology, as the operation of inversion is not continuous in this topology. Instead the ideles are identified with the closed subset of all pairs (x,y) of A×A with xy=1, with the subset topology. The idele group may be realised as the restricted product of the unit groups of the local fields with respect to the subgroup of local integral units. The ideles form a locally compact topological group.
The principal ideles are given by the diagonal embedding of the invertible elements of the number field or field of functions and the quotient of the idele group by principal ideles is the idele class group. This 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 from the idele group to the Galois group of the maximal abelian extension of the number or function field. The Artin reciprocity law, which is a high level generalization 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 from the idele class group to the abelian part of the absolute Galois group of the field.
The self-duality of the adeles of the function field of a curve over a finite field easily implies the Riemann–Roch theorem for the curve and the duality theory for the curve.
As a locally compact abelian group, the adeles have a nontrivial translation invariant measure. Similarly, the group of ideles has a nontrivial translation invariant measure using which one defines a zeta integral. The latter was explicitly introduced in papers of Kenkichi Iwasawa and John Tate. The zeta integral allows one to study several key properties of the zeta function of the number field or function field in a beautiful concise way, reducing its functional equation of meromorphic continuation to a simple application of harmonic analysis and self-duality of the adeles, see Tate's thesis.
The ring A combined with the theory of algebraic groups leads to adelic algebraic groups. For the function field of a smooth curve over a finite field the quotient of the multiplicative group (i.e. GL(1)) of its adele ring by the multiplicative group of the function field of the curve and units of integral adeles, i.e. those with integral local components, is isomorphic to the group of isomorphisms of linear bundles on the curve, and thus carries a geometric information. Replacing GL(1) by GL(n), the corresponding quotient is isomorphic to the set of isomorphism classes of n vector bundles on the curve, as was already observed by André Weil.
Another key object of number theory is automorphic representations of adelic GL(n) which are constituents of the space of square integrable complex valued functions on the quotient by GL(n) of the field. They play the central role in the Langlands correspondence which studies finite-dimensional representations of the Galois group of the field and which is one of noncommutative extensions of class field theory.
Another development of the theory is related to the Tamagawa number for an adelic linear algebraic group. This is a volume measure relating G(Q) with G(A), saying how G(Q), which is a discrete group in G(A), lies in the latter. A conjecture of André Weil was that the Tamagawa number was always 1 for a simply connected G. This arose out of Weil's modern treatment of results in the theory of quadratic forms; the proof was case-by-case and took decades, the final steps were taken by Robert Kottwitz in 1988 and V. I. Chernousov in 1989. The influence of the Tamagawa number idea was felt in the theory of arithmetic of abelian varieties through its use in the statement of the Birch and Swinnerton-Dyer conjecture, and through the Tamagawa number conjecture developed by Spencer Bloch, Kazuya Kato and many other mathematicians.
Almost any book on modern algebraic number theory, such as: