Zeta and L-functions | Algebraic number theory

Tate's thesis

In number theory, Tate's thesis is the 1950 PhD thesis of John Tate 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. (Wikipedia).

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Related pages

Analytic continuation | L-function | Emil Artin | Theta function | Functional equation | Jean Dieudonné | Erich Hecke | Class field theory | Peter Gustav Lejeune Dirichlet | Kenkichi Iwasawa | Poisson summation formula | Basic Number Theory | Harmonic analysis | Hecke character | Number theory | General linear group