Tate's thesis

Summary

Tate's thesis is the 1950 doctoral thesis of John Tate, written under Emil Artin. It proves the meromorphic continuation and the functional equation of the L-functions of Hecke characters. The tool is harmonic analysis on the adele ring and on the idele group. The L-function appears as a zeta integral of a test function against a character over the ideles. The functional equation then follows from the Poisson summation formula for the discrete subgroup $K$ of $\mathbb{A}_K$. The method made adeles a standard tool and is a model for the theory of automorphic L-functions.