Problems in the Theory of Automorprhic Forms To Salomon Bochner In Gratitude

1. There has recently been much interest, if not a tremendous amount of progress, in the arithmetic theory of automorphic forms. In this lecture I would like to present the views not of a number theorist but of a student of group representations on those of its problems that he finds most fascinating. To be more precise, I want to formulate a series of questions which the reader may, if he likes, take as conjectures. I prefer to regard them as working hypotheses. They have already led to some interesting facts. Although they have stood up for a fair length of time to the most careful scrutiny I could give, I am still not entirely easy about them. Indeed even at the beginning in the course of the definitions, which I want to make in complete generality, I am forced, for lack of time and technical competence, to make various assumptions.