Arithmetic Applications of the Trace Formula

PI在数论中提出了几个以迹公式为核心的证明研究项目。大部分项目可以分为三个部分。首先，从经典和模p/p-进朗兰兹规划的角度研究志村变的上同性。其次，PI涉及自同构表示及其l -函数有限族的算术统计量。第三，PI参与了一个联合项目，该项目旨在对具有算术应用的酉群的自同构表示进行分类。一些项目将与Kaletha， mininguez, Scholze， Templier和White合作进行。提出的项目源于古希腊人试图回答的基本问题，比如研究素数的性质，找到整数或有理数多项式方程的系统解。微量公式可以看作是20世纪的发明，它有效地解决了这些基本问题。自同构形式和朗兰兹纲领在理论物理学中变得越来越重要，可能会对我们对宇宙的理解有所启发。数论的发展在互联网和电子设备广泛使用的新时代尤为重要，因为它广泛应用于密码学，纠错码和互联网安全，仅举几例。

The PI proposes several research projects in number theory which involve the trace formula at the core of proof. Largely the projects can be divided into three parts. First, the PI proposes to study the cohomology of Shimura varieties with a view towards the classical and mod p/p-adic Langlands program. Second, the PI is concerned with arithmetic statistics for in finite families of automorphic representations and their L-functions. Third, the PI is involved in a joint project to classify the automorphic representations of unitary groups with arithmetic applications in mind. Some projects will be carried out in collaboratiton with Kaletha, Minguez, Scholze, Templier and White. The proposed projects stem from basic questions that ancient Greeks were trying to answer, such as studying properties of prime numbers and finding systematic solutions to polynomial equations in integers or rational numbers. The trace formula may be viewed as a twentieth century invention to tackle such fundamental problems effectively. Automorphic forms and the Langlands program become increasingly important in theoretical physics and would possibly shed light on our understanding of the universe. Developments in number theory are particularly important in the new era making extensive use of internet and electronic devices in view of its wide applications to cryptography, error-correcting codes and internet security, just to name a few.