Arithmetic Aspects of the Langlands Program

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Number theory is the branch of mathematics that studies phenomena related to properties of whole numbers. A typical number theoretic question is to find all the whole number solutions of some equation of interest. By synthesizing the topological and algebraic properties of the graphs of these equations, one obtains a mathematical object (a collection of mathematical data) called a Galois representation which encodes many of the properties of the originating equation. The mathematician Robert Langlands has developed a series of conjectures (or mathematical predictions) which anticipate a very close relationship between these Galois representations and some very different kinds of group representations, known as automorphic representations, which arise out of mathematical analysis. The goal of this project is to make progress on Langlands's conjectures by placing them in various more structural geometric frameworks, and then to use geometric methods and perspectives to establish relationships between these two different kinds of representations (Galois and automorphic). The project provides training opportunities for graduate students.The goal of the research project is to investigate both local and global aspects of the arithmetic Langlands program. The principal investigator, Dr. Matthew Emerton, will construct moduli stacks of global Galois representations, and study their geometric properties, as well as the properties of the global-to-local restriction maps which map them to (products of) moduli stacks of local Galois representations. These investigations are closely related to the problem of extending the Taylor--Wiles--Kisin patching method to the residually reducible case, and to the study of the Eisenstein part of cohomology, and Dr. Emerton will pursue these relationships, especially in the context of odd two-dimensional representations, which are connected to the cohomology of modular curves. Dr. Emerton will also develop a new approach to the mod p deformation theory of crystalline local-at-p Galois representations, via their relationship to prismatic F-crystals and gauges. The resulting improved understanding of this deformation theory should lead to new global results, such as new cases of the weight part of Serre's conjecture and new cases of the Breuil--Mezard conjecture.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.