Harmonic analysis on the adeles

Summary

Harmonic analysis on the adeles is Fourier analysis on the adele ring $\mathbb{A}_K$ of a number field $K$, seen as a locally compact abelian group. It rests on three facts. The group $\mathbb{A}_K$ carries a Haar measure. It is isomorphic to its own Pontryagin dual through a character that is trivial on $K$. The subgroup $K$ is discrete, and the quotient $\mathbb{A}_K / K$ is compact. These facts give a Poisson summation formula for the lattice $K$ in $\mathbb{A}_K$. This is the method of Tate's thesis, which proves the functional equations of the L-functions of Hecke characters. It is also the setting of the modern theory of automorphic forms.