Arithmetic Applications of the Trace Formula

PI提出了几个数论研究项目，其中涉及证明核心的迹公式。项目大致可分为三个部分。首先，PI建议研究Shimura簇的上同调，着眼于经典和mod p/p-adic Langlands程序。第二，PI关注有限族自守表示及其L-函数的算术统计。第三，PI参与了一个联合项目，对酉群的自守表示进行分类，并考虑到算术应用。一些项目将与Kaletha、Minguez、Scholze、Templier和白色合作进行。 提出的项目源于古希腊人试图回答的基本问题，例如研究素数的性质，并找到整数或有理数多项式方程的系统解。迹公式可以被看作是二十世纪世纪的发明，以有效地解决这些基本问题。自守形式和朗兰兹纲领在理论物理学中变得越来越重要，并可能为我们理解宇宙提供线索。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.