Tate’s Thesis on Zeta Functions on Number Fields

In this paper, we examine John Tate’s seminal work calculating functional equations for zeta functions over a number field k. Tate examines both ‘local’ properties of k, completed with respect to a given norm, and ‘global’ properties. The global theory examines the idele and adele groups of k as a way of encoding information from all of the completions of k into single structures, each with its own meaningful topology, measure, and character group. Finally, Tate uses techniques from Fourier analysis, both on the local fields and on the adele group to find functional equations for the zeta functions he defines.