Local Langlands in Families

The Langlands program is a broad set of conjectures that provide an avenue toward understanding the mysterious properties of L-functions, which encode information about solutions to polynomial equations. These conjectures have motivated broad swaths of research in the field of algebraic number theory and have shed light on long-standing open problems. In some contexts, the Langlands conjectures have been proven true, but there remain subtleties that are poorly understood. In other contexts, they are yet to even be precisely formulated. This project proposes to study the Langlands program within the context of geometric deformation. In addition to the number theory research goals, the project also includes a training component for educators: a three-week summer number theory program during each year of the project for high school teachers throughout Maine, especially those serving rural areas. As part of the program, educators will develop extracurricular activities or elective courses in number theory for their local high school. The first research goal of the project is to make progress toward establishing the so-called local Langlands conjecture in families, which upgrades a presumed local Langlands correspondence to a map between moduli spaces of integral objects on either side of the correspondence, assuming a short list of desiderata. The second goal is to study the logical dependence between the local Langlands conjecture in families and the conjectures surrounding recent p-adic geometrizations of the local Langlands conjectures. The method is to generalize converse theorems and the Plancherel measure in families in order to extend a line of argument previously used in the special cases of general linear groups and classical groups. This project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR). 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.