abelian group. The character group of is the set of all characters of and is denoted by Equivalently is the set of all continuous group homomorphisms from to We equip with the topology of uniform convergence on compact subsets of One can show that is also a locally compact abelian group. Theorem. The adele ring is self-dual: Proof. By reduction to local coordinates, it is sufficient to show each is self-dual. This can be done by using a fixed character of We illustrate this idea by showing is self-dual. Define: Then the following map is an isomorphism which respects topologies: Theorem (algebraic and continuous duals of the adele ring). Let be a non-trivial character of which is trivial on Let be a finite-dimensional vector-space over Let and be the algebraic duals of and Denote the topological dual of by and use and to indicate the natural bilinear pairings on and Then the formula for all determines an isomorphism of onto where and Moreover, if fulfils for all then Tate's thesis With the help of the characters of we can do Fourier analysis on the adele ring. John Tate in his thesis "Fourier analysis in Number Fields and Hecke Zeta Functions" 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. We can define adelic forms of these functions and we can represent them as integrals over the adele ring or the idele group, with respect to corresponding Haar measures. We can show functional equations and meromorphic continuations of these functions. For example, for all with where is the unique Haar measure on normalized such that has volume one and is extended by zero to the finite adele ring. As a result, the Riemann zeta function can be written as an integral over (a subset of) the adele ring. Automorphic forms 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: And finally where is the centre of Then we define an automorphic form as an element of In other words an automorphic form is a function