Langlands Correspondences and Motivic L-Functions

The study of the abstract properties of numbers and their relations has appeared at an early stage in the history of every civilization, and reflection on the problems of number theory is consistently found at the root of most of the ideas that characterize contemporary life, from timekeeping, to the symmetry concepts of modern physics, to the logic of computers. Numbers can be studied in two different ways that are nearly independent: they can be used for measurement and they can be used to do arithmetic. The interaction between these two properties has always been the basis of number theory. The second half of the twentieth century saw the formulation of several ambitious research programs that aimed at obtaining a systematic understanding of these interactions by studying numbers and their relations with the help of symmetry. The branch of mathematics concerned with their geometric symmetries is called arithmetic geometry; the branch concerned with their dynamical symmetries is called automorphic forms. The Langlands program aims to unify these two branches by showing how each kind of symmetry encodes the other. Contemporary theoretical physics has introduced the concept of higher order symmetries, based on new notions of space that themselves owe a great deal to earlier developments in number theory; more recently, higher order symmetries have been of increasing importance in the Langlands program. This project explores the role of higher order symmetries in connection with several specific questions in the Langlands program, with the ultimate aim of contributing to the understanding of solutions of equations in whole numbers.

The project is a contribution to the arithmetic theory of automorphic forms, in the setting of the Langlands program, with special attention to the arithmetic of motives and their associated Galois representations, directly or by application of congruence methods. The specific goals of the project are the study of the local Langlands parametrizations for general groups, using trace formula methods; the proof of Deligne's conjecture on special values of L-functions, especially tensor product L-functions; the verification of Venkatesh's conjecture on derived Hecke algebras for modular forms of weight 1, using an unexpected relation with p-adic L-functions; and the development of a character theory for mod p representations of p-adic groups. The methods involved in the present project combine standard techniques from arithmetic geometry and automorphic forms, an approach to cohomological automorphic forms based on differential geometry and representation theory, categorical representation theory, as well as new methods.