Langlands correspondences and the arithmetic of automorphic forms

Algebra is taught as the art of solving equations, but the fact is that very few kinds of algebraic equations can be solved by mechanical rules. Nevertheless, one can often characterize an equation by the shape of its set of solutions, even if the solutions themselves cannot be written down. The Langlands correspondences grow out of the observation that the shapes of the equations that arise in two apparently different areas of mathematics — number theory and the symmetries of mathematical physics — are linked by a complex web of relations that allow questions in one area to be solved by reference to the other area. The branch of mathematics that studies these correspondences is called the theory of automorphic forms. The simplest examples of automorphic forms are the familiar sine and cosine function from trigonometry. More general automorphic forms are described in terms of geometry in higher dimensions. In this way the study of automorphic forms contributes to the development of all branches of mathematics. Problems studied in the present project, presented at seminars and conferences, will serve as the basis for training the next generation of specialists. They also provide examples for philosophers and historians of the kind of synthesis of ideas that is characteristic of contemporary mathematics, and that presents a special challenge for those who prodict that artificial intelligence will play a prominent role in the mathematics of the future.This project is a contribution to the study of motives over number fields in the setting of the Langlands program, continuing a theme that has been central to the PI's research throughout his career. The present proposal is divided into two parts. Part 1 combines motivic methods, — especially the Grothendieck-Deligne theory of weights — with the Selberg trace formula and representation theory to the study of local and global Langlands correspondences, both classical and mod p. In particular, a strategy is outlined for an inductive construction of the local Langlands correspondence over local fields of positive characteristic, and an extension is proposed to Arthur parameters of the author's approach to the generalized Ramanujan conjecture. Part 2 points in the opposite direction: it applies the insights of the Langlands program and harmonic analysis on reductive groups to study (p-adic) motivic L-functions, especially square root p-adic L-functions, with special attention to classifying the Gan-Gross-Prasad periods that can be interpreted as cohomological cup products. A more speculative project, joint with Feng and Mazur, aims to provide a categorical framework for Venkatesh's motivic conjectures in the setting of Iwasawa theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.