The Arithmetic Properties of Modular Forms and Hypergeometric Systems

模形式和超几何系统的算术性质

• 批准号: 2302531
• 负责人: Fang-Ting Tu
• 金额: $ 16.46万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302531&HistoricalAwards=false
• 关键词: Arithmetic; Properties; Modular; Forms; Hypergeometric

Number theory is, essentially, the study of the properties of numbers. This seemingly simple concept leads to remarkably difficult unsolved problems in mathematics, with implications in areas such as biology, chemistry, computer science, and physics. This project focuses on investigating the connection between two fundamental objects in number theory: modular forms and hypergeometric functions. The theory of classical modular forms has long played an important role in number theory and was essential in Wiles’ proof of Fermat’s Last Theorem. More recently, generalized modular forms have become central objects of study, and can be understood through differential equations satisfied by classical modular forms. Special functions known as classical hypergeometric functions are known to satisfy very similar differential equations, suggesting a connection between hypergeometric functions and modular forms. In turn, hypergeometric functions provide arithmetic information for various mathematical objects, including multi-parameter families of Calabi-Yau manifolds leading to applications in string theory. An overall expectation is that hypergeometric functions provide a new direction in understanding the phenomena arising in mirror symmetry, one of the central research themes binding string theory and algebraic geometry. This project will make use of this connection to hypergeometric functions to study the arithmetic properties and applications of general modular forms. The broader impacts of this project include mentoring graduate and undergraduate students in research, organizing conferences and workshops, continuing to work on outreach programs with middle and high school students, and disseminating data and expository notes.Specifically, this project will study the properties of modular forms in relation to character sums and differential equations – especially those of hypergeometric type – using methods from arithmetic geometry, Galois theory, and Galois representations. The PI will focus on the exploration of modular forms on Shimura curves – including classical modular curves – which are moduli spaces of certain varieties. The main goals are: (1) to discover the arithmetic properties of modular forms on Shimura curves through explicit constructions; (2) to advance the understanding of hypergeometric systems and the modularity of hypergeometric Galois representations; and (3) to exploit the relations between hypergeometric functions and modular forms for arithmetic triangle groups to understand the fundamental properties of these two objects, such as their values at complex multiplication points and L-values.This project is jointly funded by the Algebra and Number Theory Program and the Established Program to Stimulate Competitive Research (EPSCoR).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.

从本质上讲，数论是研究数的性质的。这个看似简单的概念导致了数学中非常困难的未解决问题，涉及到生物、化学、计算机科学和物理等领域。本课题致力于研究数论中的两个基本对象之间的联系：模形式和超几何函数。经典模形式理论在数论中一直扮演着重要的角色，在Wiles对费马大定理的证明中起着至关重要的作用。最近，广义模形式已成为研究的中心对象，并且可以通过经典模形式满足的微分方程式来理解。众所周知，被称为经典超几何函数的特殊函数满足非常相似的微分方程式，这表明超几何函数和模形式之间存在联系。反过来，超几何函数为各种数学对象提供算术信息，包括导致弦理论应用的多参数Calabi-Yau流形家族。总的期望是超几何函数为理解镜像对称现象提供了一个新的方向，镜像对称是弦理论和代数几何的中心研究主题之一。本项目将利用与超几何函数的这种联系来研究一般模形式的算术性质和应用。这个项目的更广泛的影响包括指导研究生和本科生的研究，组织会议和研讨会，继续开展与初中生和高中生的外展项目，以及传播数据和说明性笔记。具体地说，这个项目将使用算术几何、伽罗瓦理论和伽罗瓦表示的方法来研究与特征和微分方程相关的模形式的性质，特别是那些超几何类型的模形式。PI将集中于探索Shimura曲线上的模形式-包括经典的模曲线-这是某些变种的模空间。主要目的是：(1)通过显式构造发现Shimura曲线上模形式的算术性质：(2)促进对超几何系统和超几何Galois表示的模性的理解；以及(3)利用超几何函数和算术三角形群的模形式之间的关系，以了解这两个对象的基本性质，如它们在复乘点处的值和L值。该项目由代数和数论计划和既定的激励竞争研究计划(EPSCoR)共同资助。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智力优势和更广泛的影响审查标准进行评估，被认为值得支持。