on satisfying certain algebraic and analytic conditions. For studying automorphic forms, it is important to know the representations of the group It is also possible to study automorphic L-functions, which can be described as integrals over We could generalize even further by replacing with a number field and with an arbitrary reductive algebraic group. Further applications A generalisation of Artin reciprocity law leads to the connection of representations of and of Galois representations of (Langlands program). 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 sweeping 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. Notes What problem do the adeles solve? Some good books on adeles References Sources 366 pages. 595 pages. 294 pages. 250 pages. Algebraic number theory Topological algebra