Moduli spaces of Galois representations

Northwestern University, Evanston IL

Investigators

Abstract

Number Theory is the branch of mathematics which studies the properties and patterns of whole numbers. Despite this seemingly elementary premise, number theory has been at the the forefront of some of the most intricate structures discovered in mathematics, as well as underlying key practical applications (such as public key cryptography, which powers current secure communications over the internet). One fundamental idea of modern number theory is that collections of numbers sharing some common feature (such as being solutions of some list of equations) possess interesting emergent properties and symmetries. The most primordial of such emergent symmetry is the absolute Galois group of the rational numbers, and a large swath of number theory in the last few centuries concerns probing its (complicated) internal structure. In the 1970s, Langlands made a web of surprising predictions that this absolute Galois group is related to the (continuous) symmetry of vibrations on some highly symmetric geometric shapes (the automorphic representations). Such conjectures are known to have far reaching consequences: for instance, a proven special case was at the heart of the resolution of Fermat's Last Theorem. One promising approach to Langlands Conjectures that crystallized over the last few decades is the method of p-adic deformation, where one organizes the information on the two sides of the conjecture according to divisibility by powers of a given prime number p. The key point whose importance has only come into focus very recently is that this process reveals macroscopic/geometric features which make it easier to match the two sides, and the project aims to study exactly those features. Belonging to an emerging research direction, the project is a fertile ground for the discovery of and experimentation with new concrete phenomena, and thus create excellent opportunities for the training of students at both the graduate and undergraduate level. The PI also plans to disseminate the new geometric perspectives in the Langlands program to a broader audience through organizing summer schools and mini-courses. More specifically, the project studies the geometry of the moduli stack of representations of the Galois groups of p-adic fields, with focus on loci cut out by p-adic Hodge-theoretic conditions. These recently constructed spaces are expected to play a pivotal role in the nascent categorical p-adic Langlands program, which seeks to promote the (conjectural) relationship between individual smooth representations of p-adic Lie groups and individual local p-adic Galois representations to a relationship between the entire categories of such objects. The project aims to establish a bridge between these two categories, by relating both to categories of sheaves on some intermediate objects, moduli spaces of (semi-)linear algebraic objects, which are susceptible to analysis via methods of geometric representation theory. A sufficiently strong control on the geometry would lead to major progress on local questions such as the Breuil-Mezard conjecture as well as global questions such as Serre weight conjectures, automorphy lifting and the structure of mod p cohomology of locally symmetric spaces. The flow of information can also be reversed, namely one can predict new phenomena in geometric representation theory from arguments and heuristics with Galois representations. Furthermore, these linear algebraic moduli spaces are sufficiently concrete that one can experiment on them with computer algebra software, leading to many theoretical and computational projects accessible to undergraduate students. 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.