Student Number Theory:Tate’s thesis

Tate’s 1950 thesis provides a deeper understanding of the functional equation of Hecke L-functions through harmonic analysis on adèles and idèles. We begin by examining Hecke’s original proof of the functional equation. We will then delve into the key components of Tate’s thesis, including local and global additive duality, and utilize these concepts to establish the functional equation and compute the root numbers in both local and global cases. Additionally, we will discuss Godement and Jacquet’s extension of Tate’s work to automorphic L-functions for GL(n).