The Trace Formula Method and the Arithmetic and Geometry of Modular Varieties in the Langlands Program

Columbia University, New York NY

Investigators

Abstract

Number theory is one of the oldest branches of mathematics. It studies properties of the integers. The impact of research in number theory on modern science and technology is significant. Most notably, number theory has proved indispensable in cryptography, internet security, telecommunication, and so on. This is a project to study the so-called Langlands program, which predicts deep relations between number theory and other seemingly unrelated branches of mathematics. Progress in the Langlands program will not only advance our knowledge in number theory, but also demonstrate the unification of mathematics, showing that seemingly different areas in mathematics are governed by certain common principles. This will be beneficial for enhancing communication and collaboration between researchers working in different branches of mathematics and will have potential applications to cryptography, internet security, telecommunication, and so on, as mentioned above. In more detail, a central topic in the Langlands program is the reciprocity between motives and automorphic forms. One of the most important tools is the trace formula. In the context of automorphic forms, there are trace formulas of Arthur-Selberg type and the so-called relative trace formulas. In the context of algebraic geometry, there are trace formulas of Grothendieck-Lefschetz-Verdier type. The investigator studies these trace formulas in conjunction with various geometric objects that naturally arise in the Langlands program. These geometric objects, which include Shimura varieties, moduli spaces of shtukas, Rapoport-Zink spaces, affine Deligne-Lusztig varieties, etc., are themselves moduli spaces of certain geometric structures. The goal of this project is to describe, in various concrete settings, the interaction between trace formulas and the arithmetic and geometry of these moduli spaces. Such an understanding will lead to results on the relation between motives and automorphic forms. 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.