P-adic Aspects of the Langlands Program

Number theory is the branch of mathematics that studies phenomena related to properties of numbers, especially the relationship between integers (whole numbers) and interesting irrational numbers (such as the square root of two, which was the first irrational number to be discovered). Surprisingly, number theoretic phenomena are sometimes governed by functions arising from a different area of mathematics, known as harmonic analysis.  This is the field that studies waves and related periodic phenomena. It turns out that problems in number theory sometimes admit solutions controlled by functions known as automorphic forms, which are periodic functions, but whose group of periods is generally non-commutative. The study of this relationship between number theory and harmonic analysis is known as the "Langlands program." The Principal Investigator will study and further develop a particular aspect of the Langlands program, the so-called p-adic Langlands program. Here p stands for a prime number, and p-adic indicates that one focuses on divisibility properties with respect to this prime. The p-adic Langlands program combines the study of harmonic analysis with p-adic methods involving algebra and geometry, with the aim of making fundamental new progress in number theory. This project will also provide research training opportunities for graduate students.

The goal of the project is to investigate the p-adic aspects of the Langlands program from a range of different viewpoints.  A major focus is the construction of a categorical p-adic local Langlands correspondence in contexts where it is not currently known to exist, by combining methods from representation theory, the geometry of moduli stacks, and the theory of differential graded algebras. The p-adic local Langlands correspondence has proved to be one of the most powerful tools available for establishing relationships between questions in algebraic number theory and the theory of automorphic forms, and so extending its known range of validity is one of the fundamental problems in number theory. Other aspects of the project include the p-adic interpolation of the trace formula, the study of prismatic cohomology of Hilbert modular varieties, and the study of global-to-local restriction maps between moduli stacks of Langlands parameters.  A common element in all these investigations is the combining of algebraic, geometric, and representation-theoretic techniques, and the project also serves to develop new methods in, and syntheses of, these areas of mathematics.

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.

Award Number: 2502609
Principal Investigator: Matthew Emerton
Funds Obligated: $210,000
State: IL