Applications
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 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.
Read more about this topic: Adele Ring