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

朗兰兹纲领中的迹公式法与模簇的算术和几何

• 批准号: 2132670
• 负责人: Yihang ZHU
• 金额: $ 12.5万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-10-01 至 2022-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2132670&HistoricalAwards=false
• 关键词: Trace; Formula; Method; Arithmetic; Geometry

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.

数论是数学最古老的分支之一。它研究整数的性质。数论研究对现代科学技术的影响是巨大的。最值得注意的是，数论已被证明在密码学、互联网安全、电信等领域不可或缺。这是一个研究所谓朗兰兹程序的项目，它预测了数论和其他看似无关的数学分支之间的深层关系。朗兰兹纲领的进展不仅将提高我们在数论方面的知识，而且还将证明数学的统一性，表明数学中看似不同的领域受到某些共同原则的支配。这将有利于加强在不同数学分支工作的研究人员之间的交流和合作，并将在密码学，互联网安全，电信等领域具有潜在的应用，如前所述。更详细地说，朗兰兹纲领的一个中心主题是动机和自同构形式之间的相互关系。最重要的工具之一是迹公式。在自同态形式中，有Arthur-Selberg型的迹公式和所谓的相对迹公式。在代数几何的背景下，存在Grothendieck-Lefschetz-Verdier型的迹公式。研究者将这些迹公式与朗兰兹程序中自然产生的各种几何对象结合起来研究。这些几何对象包括Shimura变体、shtukas的模空间、Rapoport-Zink空间、仿射delign - lusztig变体等，它们本身就是某些几何结构的模空间。这个项目的目标是描述，在各种具体设置中，迹公式与这些模空间的算术和几何之间的相互作用。这样的理解将导致动机和自同构形式之间关系的结果。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。