Arithmetic of automorphic L-functions

In this work, working exclusively with GL(2) over Q, I will introduce the three main ingredients in understanding special values: (i) DEfinition of certain periods attached to automorphic forms; these periods arise via a comparison of rational structures on Whittaker models and on cuspidal cohomology; (ii) Relations amongst these periods; and (iii) Interpretting the classical Hecke-Mellin transform of a modular form as a Poincare duality pairing. The entire discussion will be in the language of automorphic representations, and will reprove classical results of Manin and Shimura on the critical values of L-functions of modular forms. (Pre-requisites: Bump's book, and some sheaf-cohomology. Reference: My paper with Naomi Tanabe titled `Notes on the arithmetic of Hilbert modular forms.')