Rational Structures on Automorphic Representations and an Automorphic Analogue of Deligne's Conjecture

It is well known that the space of modular forms as Hecke module comes with a canonical rational structure which, by classical work of Manin and Shimura, explains the rationality of special values of L-functions attached to cusp forms. In this talk we will discuss how this result extends to automorphic L-functions attached to groups of higher rank.