Tate's thesis

In number theory, Tate's thesis is the 1950 PhD thesis of John Tate (1950) completed under the supervision of Emil Artin at Princeton University. In it, Tate used a translation invariant integration on the locally compact group of ideles to lift the zeta function twisted by a Hecke character, i.e. a Hecke L-function, of a number field to a zeta integral and study its properties. Using harmonic analysis, more precisely the Poisson summation formula, he proved the functional equation and meromorphic continuation of the zeta integral and the Hecke L-function. He also located the poles of the twisted zeta function. His work can be viewed as an elegant and powerful reformulation of a work of Erich Hecke on the proof of the functional equation of the Hecke L-function. Erich Hecke used a generalized theta series associated to an algebraic number field and a lattice in its ring of integers.