The Arithmetic Properties of Modular Forms and Hypergeometric Systems

数论本质上是对数字性质的研究。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，并在生物学、化学、计算机科学和物理学等领域产生了影响。这个项目的重点是研究数论中两个基本对象之间的联系：模形式和超几何函数。经典模形式理论长期以来在数论中占有重要地位，在怀尔斯证明费马大定理的过程中发挥了重要作用。最近，广义模形式已经成为研究的中心对象，并且可以通过满足经典模形式的微分方程来理解。被称为经典超几何函数的特殊函数已知满足非常相似的微分方程，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了算术信息，包括导致弦理论应用的多参数Calabi-Yau流形族。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核心研究主题之一。本课题将利用这种与超几何函数的联系来研究一般模形式的算术性质和应用。该项目的广泛影响包括指导研究生和本科生进行研究，组织会议和研讨会，继续开展面向初高中学生的推广项目，以及传播数据和说明性笔记。具体来说，该项目将使用算术几何、伽罗瓦理论和伽罗瓦表示的方法，研究与特征和和微分方程（特别是超几何型的微分方程）相关的模形式的性质。PI将专注于探索志村曲线上的模形式——包括经典的模曲线——这是特定种类的模空间。主要目标是：(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伽罗瓦表示的模块化的理解；(3)利用算术三角群的超几何函数和模形式之间的关系，了解这两个对象的基本性质，例如它们在复乘法点和l值处的值。该项目由代数和数论项目和促进竞争研究的既定项目（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。

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.