Functoriality in the Mod-p Langlands Program

Number theory is the branch of mathematics that deals with properties of whole numbers and whole number solutions to polynomial equations, and stands as one of the oldest mathematical disciplines. Representation theory, another equally influential branch of mathematics, quantifies symmetries of geometric objects (such as a square or a hydrogen atom), and has important uses in physics.

Though seemingly unrelated, these two areas are intimately linked by the Langlands Program, a vast set of conjectures that allows for the transfer of results and theorems between number theory and representation theory. It is of paramount importance to understand these conjectures, since tools from one discipline can be imported to tackle previously intractable problems in another (the proof of Fermat's Last Theorem being a prime example).

This has pushed the Langlands Program to the forefront of current research. The present project seeks to establish instances of a local version of the Langlands Program with mod p coefficients, so that information from representation theory can be transferred into arithmetic data.

The setting of the current project lies within the representation theory of p-adic reductive groups (such as GL_2(Q_p)) on mod p vector spaces. Such representations are exceedingly intricate, and one of the main goals is to use derived categories in order to more precisely relate such representations to modules over differential graded Hecke algebras.

This will allow for the use of new tools to understand the relationships between Langlands correspondences for varying groups. In addition to this, the PI and his collaborators plan to use known instances of automorphic base change and the global theory of automorphic forms to develop a mod p Langlands correspondence for p-adic unitary groups. This would enrich the known instances of mod p Langlands correspondences by showing that they are compatible with functorial constructions.

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.