Fourier Analysis in Number Fields

Abstract

In this thesis we give an exposition of John Tate's doctoral dissertation titled `Fourier Analysis in Number Fields and Hecke's Zeta-Functions'. In this dissertation, Tate proved the analytic continuation and functional equation for Hecke's -function over a number eld k using what is now known as harmonic analysis over ad eles. In his work he rst examines the local -function and then uses ad eles and id eles to include in a symmetric way all the completions of the eld into a single structure, so as to examine the global -function. We explain required prerequisites and expand upon ideas used in Tate's thesis to give a comprehensive view of Tate's work.