L-functions of Hecke Characters and Cohomology

Abstract

John Tate in his doctoral dissertation, “Fourier Analysis on Number Fields and Hecke’s Zeta Function”, established the meromorphic continuation and functional equation of Hecke’s Zeta Function over a number field using methods of harmonic analysis on the ad`ele ring of the number field. The theory in Tate’s thesis can be extended to L-functions that are attached to Hecke characters - which are id`ele class group characters. In this thesis, we study the necessary background and explore the key concepts to provide a comprehensive exposition of Tate’s work. Further, we continue to study Hecke characters - the associated L-functions along with the arithmetic aspects of these L-functions.