The Adelic point of view on Abelian L-functions

In this thesis we present the Iwasawa-Tate adelic framework which leads to the proof of the analytic continuation of Hecke L-functions and their functional equation. We follow the representation theory approach provided by Kudla's exposition of Tate's thesis in ‘An Introduction to the LANGLANDS PROGRAM’, developing the details of the results therein. In this context, the focus is on a representation of the idèles in the space of tempered distributions on the adèles, and its analogue for local fields. Tate's zeta integrals are then interpreted in terms of the sub-representations associated with idele class characters.