The Arithmetic of Hypergeometric Varieties and Noncongruence Modular Forms

数论是现代数字技术的科学支柱之一，在编码理论和密码学中有着重要的应用。从20世纪90年代费马大定理的证明开始，数论进入了一个新时代，融合了其他学科的丰富思想和工具，拥抱了快速发展的数学软件包和数据库，并建立了多层平台和社区进行合作。这个研究项目调查模形式，这是数论中的中心对象，贯穿数学和物理学。这项工作旨在扩大和加深对这一基本领域的理解。研究人员计划通过继续指导初级数学家和组织国家和国际研讨会和会议来扩大研究计划的影响。 本项目研究超几何簇和非全等模形式的运算。大多数模形式是非同余的，因为它们的对称群不能用同余来描述。 这两个主题之间的联系的研究是明显的事实，即所有代数曲线定义的一些领域可以实现为模块曲线的有限指数子群的模块组。模形式是模曲线的算术不变量。在所有的代数曲线中，广义Legendre曲线是理论和计算研究中特别友好的模型。超几何簇是广义Legendre曲线的高维类似物。本课题的主要目的是：（1）将一类经典超几何公式转换到有限域上，（2）利用这些转换公式计算超几何簇的算术不变量，（3）基于超几何簇的显式Galois表示，得到与非同余模形式相关的Galois表示的新的自同构结果;（4）进一步研究了非同余模形式的其他性质，包括一个长期存在的无界分母猜想.

Number theory is one of the scientific backbones of modern digital technology and has important applications in coding theory and cryptography. Beginning with the proof of Fermat's Last Theorem in the 1990's, number theory entered a new era, incorporating rich ideas and tools from other subjects, embracing rapidly-developing mathematical software packages and databases, and building multi-layer platforms and communities for collaborations. This research project investigates modular forms, which are central objects in number theory that occur throughout mathematics and physics. The work aims to broaden and deepen understanding in this fundamental area. The investigator plans to widen the impacts of the research program by continued mentoring of junior mathematicians and organization of national and international workshops and conferences. This project investigates the arithmetic of hypergeometric varieties and noncongruence modular forms. A majority of modular forms are noncongruence in the sense that their symmetry groups cannot be described in terms of congruences. The link between the two topics of study is manifest by the fact that all algebraic curves defined over number fields can be realized as modular curves of finite index subgroups of the modular group. Modular forms are arithmetic invariants of the modular curves. Among all algebraic curves, the generalized Legendre curves are particularly amiable models for both theoretical and computational investigations. Hypergeometric varieties are higher dimensional analogues of the generalized Legendre curves. The objectives of this research project are: (1) to translate a class of classical hypergeometric formulas to the finite field setting; (2) to use these transformation formulas to compute arithmetic invariants of the hypergeometric varieties; (3) to obtain new automorphy results for Galois representations associated with noncongruence modular forms based on explicit Galois representations arising from hypergeometric varieties; and (4) to further investigate other properties of noncongruence modular forms, including a longstanding unbounded denominator conjecture.