Automorphic Representations and Number Theory *

An excellent introduction to the theory of automorphic representations and the relations with number theory is the Corvallis proceedings, Automorphic Forms, Representations and L-Functions, Parts 1 and 2, Proc. Sympos. Pure Math., vol. 33, 1979. Although composed mainly of survey articles, the proceedings are already rather formidable. They are a measure of the breadth of the field. They will be most useful to mathematicians who are already experts in some branch of the subject. Our purpose here is to give a modest introduction to the Corvallis proceedings. More precisely, our goal is to describe the Langlands functoriality conjecture, a mathematical insight of great beauty and simplicity. We will try to show both why it is a compelling question, and how it arose historically from Langlands' work on Eisenstein series. We hope that mathematicians from diverse or at least neighboring fields will find these notes accessible and will be encouraged to read other survey articles [2], [6], or to plunge directly into the Corvallis proceedings. * Lectures given at the Canadian Mathematical Society Summer Seminar, Harmonic Analysis, McGill University, Aug. 4-22, 1980. Copyright © 1981, American Mathematical Society