The Arithmetic Properties of Modular Forms and Hypergeometric Systems

数论本质上是研究数的性质。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，在生物学，化学，计算机科学和物理学等领域都有影响。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扮演着重要的角色，并且在怀尔斯证明费马大定理时至关重要。最近，广义模形式已经成为研究的中心对象，并且可以通过经典模形式所满足的微分方程来理解。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的联系。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卡-丘流形族。一个总体的期望是，超几何函数提供了一个新的方向，在理解镜像对称现象，其中一个中心的研究主题绑定弦理论和代数几何。本计画将利用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。该项目的更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，继续与初中和高中学生开展外联方案，以及传播数据和简要说明。本专题将研究模形式与特征和及微分方程的关系，特别是超几何类型的模形式，运用算术几何、伽罗瓦理论和伽罗瓦表示的方法。PI将专注于探索志村曲线上的模形式-包括经典的模曲线-这是某些种类的模空间。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）加深对超几何系统和超几何Galois表示的模性的理解;（3）利用超几何函数与算术三角形群的模形式之间的关系，理解这两个对象的基本性质，例如它们在复数乘法点和L值的值。该项目由代数和数论计划和刺激竞争研究的既定计划（EPSCoR）共同资助该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的评估被认为值得支持。影响审查标准。

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.