Shimura Varieties and Automorphic Forms with Arithmetic Applications

点击翻译按钮获取中文摘要

Number theory studies integers, prime numbers, and solutions of an equation in integers or rational numbers. In the digital age, number theory has been essential in algorithms, cryptography, and data security. Modern mathematics has seen increasingly more interactions between number theory and other areas from a unifying perspective. A primary example is the Langlands program, comprising a vast web of conjectures and open-ended questions. Even partial progress has led to striking consequences such as verification of Fermat's Last Theorem and resolution of the Sato-Tate conjecture, the Serre conjecture, and more. This project aims to broaden understanding of the Langlands program and related questions.The research focuses on the following topics: (1) new instances of the global Langlands correspondence for the groups GSp(2n), GSO(2n), and SO(2n); (2) a full stable trace formula for Igusa varieties and applications; (3) local harmonic analysis questions on supercuspidal representations such as invariant dimensions; and (4) a completion of the endoscopic classification for non-quasi-split unitary groups. The results of this research will stimulate further progress and inspire new investigations. Graduate students will be supported to take part in these projects.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.