CAREER: Quantifying congruences between modular forms

数论是研究最基本的数学对象--整数。由于整数是如此基本，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。人们可以考虑由方程的图形给出的形状，或者方程具有的一组对称性，或者其系数来自于对各种数系上的解的数目进行计数的函数。这个形状的几何性质、这些对称的代数性质以及这个函数的分析性质都与方程的整数解的行为密切相关。数字理论家使用这些数学领域中的每一个领域的技术，但在这个过程中，也发现了这些领域之间令人惊讶的联系，一个领域的发现可以导致另一个领域的增长。在数论中，几何、代数和分析之间的联系特别紧密的一个部分是模形式领域。所提出的研究集中在不同模形式之间的一种重要且众所周知的关系类型，称为同余，其目的是计算与给定模形式同余的形式的数目，并揭示这种计算的数论意义。推动这一项目的许多猜测都是通过实验和计算机计算找到的。主要的教育目标是帮助培训下一代理论数学家的计算和实验方法。为了实现这一点，首席调查员(PI)将为各种本科生代数和数论课程设计软件模块，这些课程提供计算实践经验。此外，PI将指导本科生进行计算研究实验，这些实验旨在数值验证项目中的猜想并探索新的方向。模形式之间的同余提供了数论中两种非常不同类型的对象之间的联系：代数对象，如伽罗瓦表示，以及分析对象，如L函数。这种联系被用作证明现代数论中一些最著名的结果的工具，例如岩泽理论的主要猜想。这项拟议的研究通过计算同余的数量，而不仅仅是确定何时存在同余，从而将同余的研究推向了一个新的、定量的方向。中心假设是，与主要猜想及其推广(如Bloch-Kato猜想)相比，同余的这种量化结构包含了关于所涉及的代数量和解析量的更精细的信息。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影响审查标准进行评估，被认为值得支持。

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.