CAREER: Quantifying congruences between modular forms

数论是研究最基本的数学对象，整数。因为整数是如此基本，数论与数学的所有主要领域都有联系。例如，考虑寻找给定方程的整数解的问题。我们可以考虑方程的图形所给出的形状，或者方程所具有的对称性集合，或者其系数来自于计算各种数系上的解的数量的函数。该形状的几何性质、这些对称性的代数性质以及该函数的分析性质都与方程的整数解的行为密切相关。数论学家使用这些数学领域的技术，但在这个过程中，也发现了这些领域之间令人惊讶的联系，即一个领域的发现可以导致另一个领域的增长。数论的一个部分，几何，代数和分析之间的联系特别强，是在模形式领域。拟议的研究集中在一个重要的和众所周知的类型之间的关系不同的模块形式称为同余，并旨在计算的形式是全等的一个给定的模块形式的数量，并揭示这种计算的数论意义。许多驱动这个项目的理论都是通过计算机计算实验发现的。主要的教育目标是促进下一代理论数学家在计算和实验方法的培训。为了实现这一目标，首席研究员（PI）将为各种本科代数和数论课程设计软件模块，提供计算实践经验。此外，PI还将指导本科生进行计算研究实验，旨在数值验证项目中的成果，并探索新的方向。模形式之间的同余关系提供了数论中两种非常不同类型对象之间的联系：代数对象，如伽罗瓦表示，和解析对象，如L函数。这个链接已被用来作为一种工具来证明一些最著名的结果在现代数论，如主要猜想的岩泽理论。该研究通过计算同余的数量，而不仅仅是确定同余何时存在，将同余的研究推向了一个新的定量方向。中心假设是，这种同余的定量结构包含了比主猜想及其推广（如布洛赫-加藤猜想）所能提供的更精细的有关代数和分析量的信息。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

Number theory is the study of the most basic mathematical objects, whole numbers. Because whole numbers are so fundamental, number theory has connections with all major areas of mathematics. For instance, consider the problem of finding the whole-number solutions to a given equation. One can consider the shape given by the graph of that equation, or the set of symmetries that the equation has, or the function whose coefficients come from counting the number of solutions over a variety of number systems. The geometric properties of this shape, the algebraic properties of these symmetries, and the analytic properties of this function are all intimately related to the behavior of the equation’s whole-number solutions. Number theorists use techniques from each of these mathematical areas, but also, in the process, uncover surprising connections between the areas whereby discoveries in one area can lead to growth in another. One part of number theory where the connections between geometry, algebra, and analysis are particularly strong is in the field of modular forms. The proposed research focuses on an important and well-known type of relation between different modular forms called congruence and aims to compute the number of forms that are congruent to a given modular form and uncover the number-theoretic significance of this computation. Many of the conjectures that drive this project were found experimentally, through computer calculations. The main educational objective is to contribute to the training of the next generation of theoretical mathematicians in computational and experimental methods. To achieve this, the Principal Investigator (PI) will design software modules for a variety of undergraduate algebra and number theory courses that provide hands-on experience with computation. In addition, the PI will supervise undergraduates in computational research experiments designed to numerically verify conjectures made in the project and to explore new directions.Congruences between modular forms provide a link between two very different types of objects in number theory: algebraic objects, like Galois representations, and analytic objects, like L-functions. This link has been used as a tool for proving some of the most celebrated results in modern number theory, such as the Main Conjecture of Iwasawa theory. The proposed research pushes the study of congruences in a new, quantitative direction by counting the number of congruences, not just determining when a congruence exists. The central hypothesis is that this quantitative structure of congruences contains finer information about the algebraic and analytic quantities involved than the Main Conjecture and its generalizations (such as the Bloch—Kato conjecture) can provide.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.