Motives and Automorphic Representations

Abstract: This paper explores relations between two separate worlds. They are the algebraic geometry of Alexander Grothendieck and the automorphic representation theory of Robert Langlands. The relation between them would be a very broad example of the fundamental duality between geometric objects and spectral objects that permeates much of modern mathematics. Our goal is to introduce explicit conjectural constructions of the universal groups that govern much of these theories, and to explore the relations between them.