Local-Global Principles in Arithmetic

Columbia University, New York NY

Investigators

Abstract

Number theory is one of the oldest branches of mathematics, and yet it continues to have more and more applications within the sciences. In this project, the principal investigator (PI) will investigate the relationship between two of the main focuses of number theory, both of which are utilized in computer science as well as physics: (1) prime numbers and the divisibility of integers and (2) algebraic solutions to polynomial equations. The fundamental idea is to understand the extent to which global objects can be arithmetically determined by the collection of its local pieces. The strategies and techniques that will be utilized in this project originate in a broad range of other mathematical subjects, including analysis, geometry, algebra and in some cases, statistics. Some of the specific questions the PI is interested in are at a level accessible to undergraduate and high-school students, and throughout the course of the project, the PI plans to utilize this to continue in educational efforts supporting an increase in diversity within mathematics. This project surrounds the widespread phenomenon of local-global principles throughout algebraic and analytic number theory, ranging from understanding obstructions of unique prime factorization in rings of integers to determining the asymptotic number of global fields with fixed invariants via the number of local extensions with fixed p-adic invariants to proving local-global compatibility results within the Langlands program. First, the PI will conduct research that furthers the statistical study of class groups that originated with the Cohen-Lenstra heuristics; amongst others, this will include proving asymptotics for class groups of families of orders. Second, the PI will study number field distributions and the local-global principles that can control their asymptotics, beginning with the case of octic quaternion number fields. The strategy for obtaining such results will rely on arithmetic invariant theory, sieve methods, and geometry-of-numbers techniques utilized frequently in the field of arithmetic statistics. On the automorphic side, the PI will investigate arithmetic and geometric properties of p-adic families of Galois representations arising from non-conjugate self-dual regular algebraic automorphic representations of the general linear group over CM fields. This will involve studying eigenvarieties and strengthening p-adic interpolation methods. 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.