FRG: Collaborative Research: Geometric Structures in the p-Adic Langlands Program

Number theory is the branch of mathematics that studies phenomena related to properties of whole numbers. A typical number theoretic question is to determine the number of whole or rational number solutions of some equation of interest. (For example, the lengths of the three sides of a right triangle are related by the Pythagorean theorem. While it is straightforward to find all right triangles whose side lengths are rational numbers, it perhaps surprisingly remains an unsolved problem to determine which whole numbers can be the area of a right triangle with rational sides.) The answers to such questions can often be encoded in certain mathematical functions known as L-functions. The mathematician Robert Langlands has developed a series of conjectures (or mathematical predictions) regarding L-functions, which predict that any L-function should arise from another kind of mathematical function called an automorphic form. One approach to the study of automorphic forms and L-functions is the use of p-adic methods. These are methods that involve using divisibility properties with respect to some fixed prime number p to study automorphic forms and L-functions. Recently, p-adic methods have begun to be unified with Langlands's ideas into a so-called "p-adic Langlands program." This project aims to develop new results and methods in the p-adic Langlands program, primarily of a geometric nature, and to use them to establish new instances of Langlands's conjectures. The award will support the training of students in this area of research that is considered of high interest. This project addresses the following fundamental question: what are the underlying geometric structures relating p-adic Galois representations to the mod p representation theory of p-adic groups? The project builds on several recent developments in which the various PIs have played key roles, including the construction of moduli stacks parametrizing p-adic representations of the Galois groups of p-adic local fields and of local models for these stacks, and recent extensions of the Taylor-Wiles patching method which relate it to the study of coherent sheaves on the local models, and to derived algebraic geometry. Some specific questions that the PIs will study are the problem of potentially crystalline lifts, the construction of a general p-adic local Langlands correspondence, and the possible local nature of the (a priori global) patching constuction. More generally, the PIs intend to introduce algebro-geometric, categorical, and derived perspectives into the p-adic Langlands program, with the intention of gaining new insights into and making new progress on some of the key open problems in the field. 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.