An introduction to Tate's Thesis.

摘要

In the early 20th century, Erich Hecke attempted to find a further generalization of the Dirichlet L-series and the Dedekind zeta function. In 1920, he [14] introduced the notion of a Grossencharakter, an ideal class character of a number field, and established the analytic continuation and functional equation of its associated L-series, the Hecke L-series. In 1950, John Tate [27], following the suggestion of his advisor, Emil Artin, recast Hecke's work. Tate provided a more elegant proof of the functional equation of the Hecke L-series by using Fourier analysis on the adeles and employing a reformulation of the Grossencharakter in terms of a character on the ideles. Tate's work now is generally understood as the GL(1) case of automorphic forms [2]. The thesis provides a thorough analysis of the approach taken by Tate in his own thesis. Background information is furnished by theory concerning topological groups, Pontryagin duality, the restricted-direct topology, and the adeles and ideles.