Collaborative Research: Slopes of Modular Forms and Moduli Stacks of Galois Representations

合作研究：伽罗瓦表示的模形式和模栈的斜率

• 批准号: 2302284
• 负责人: John Bergdall
• 金额: $ 16.27万
• 依托单位: University of Arkansas
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302284&HistoricalAwards=false
• 关键词: Collaborative; Research; Slopes; Modular; Forms

Whole numbers are among the most practical and most important mathematical objects. Humans have studied them for millennia. Number theory aims to understand patterns possessed by whole numbers. Fundamental questions revolve around multiplication: how often are numbers in some sequence even (i.e. divisible by two)? Divisible by three? Or five? Nineteenth century researchers introduced symmetry actions to reveal hidden patterns in numbers. And, in the 1970's, Robert Langlands made far-reaching conjectures on symmetry. These conjectures have occupied number theorists ever since. They predict patterns seen by symmetry actions will arise equally from the calculus of complex numbers ("modular forms"). A pattern appearing in two places is an example of a mathematical reciprocity. This project will refine Langlands' reciprocity prediction. The new tool is geometric spaces of symmetry actions, constructed by Emerton and Gee over the past fifteen years. These spaces are believed to convert reciprocity questions into geometrical ones. This project establishes instances of this belief. It will connect divisibility patterns from the world of modular forms to geometrical theorems on Emerton and Gee's spaces. The project has substantial broader impacts. Computational data will be included in the widely-used L-functions and Modular Forms Database. The project also develops computational tools for teaching. Open education resources (OERs) are learning materials placed in the public domain. Their primary benefit is providing learning experiences at low costs. They can be adapted to fit a diversity of learning environments. The project develops OERs for computer-based learning of number theory and abstract algebra. The project supports education and outreach in two more ways. First, Math Circles will be run in public schools. Second, research projects will be developed to support the Program in Mathematics for Young Scientists. Finally, the project plans two research workshops in number theory. Both aim to disseminate new advances in number theory and reciprocity. The more detailed aim is a new study of p-adic slopes of modular forms and Galois representations. The p-adic slope of a modular form is how often its p-th Hecke eigenvalue is divisible by a fixed prime p. Predictions and theorems on slopes have been around since the 1980's. Seven years ago, Bergdall and Pollack proposed a way ("the ghost conjecture") to unify almost all prior ideas. The ghost conjecture's input is a congruence class of modular forms. The output is an elementary recipe for slopes in the class. The main caveat is the ghost conjecture only applies to "regular" classes. But, assuming regularity, Liu, Truong, Xiao, and Zhao (LTXZ) recently established the conjecture. The current project removes the regular assumption in the ghost conjecture. The new tool is Emerton and Gee's (EG) moduli stack of Galois representations. In Galois terms, regularity is a generic property on the EG stack. The project's technical innovation is thus deforming slope questions over the stack. A geometrical reformulation will open the door to generalizing the LTXZ proof. It will also create space for novel studies of Hilbert modular forms or higher rank automorphic forms.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.

整数是最实用，最重要的数学对象之一。人类已经研究了数千年。数字理论旨在理解全数字所拥有的模式。基本问题围绕乘法旋转：甚至是某个顺序的数字多久（即可除以两个）？可除以三个吗？还是五个？十九世纪的研究人员提出了对称行动，以揭示数字中隐藏的模式。而且，在1970年代，罗伯特·兰兰兹（Robert Langlands）对对称性做出了深远的猜想。从那以后，这些猜想一直占领了数字理论家。他们预测对称作用看到的模式将同样来自复数的计算（“模块化表单”）。在两个地方出现的模式是数学互惠的一个示例。该项目将完善兰兰兹的互惠预测。新工具是对称作用的几何空间，在过去的15年中，由Emerton和Gee构建。据信这些空间将互惠问题转化为几何问题。该项目建立了这种信念的实例。它将从模块化形式的世界连接到埃默顿和Gee空间的几何定理。该项目具有更广泛的影响。计算数据将包括在广泛使用的L功能和模块化表单数据库中。该项目还开发了教学计算工具。开放教育资源（OER）是学习材料，放置在公共领域中。他们的主要好处是以低成本提供学习经验。它们可以适应各种学习环境。该项目开发了基于计算机的数字理论学习和抽象代数的OER。该项目通过另外两种方式支持教育和外展。首先，数学界将在公立学校进行。其次，将开发研究项目，以支持年轻科学家的数学计划。最后，该项目计划了两个研究研讨会。两者都旨在传播数量理论和互惠方面的新进步。更详细的目的是对模块化形式和GALOIS表示的P-Adic斜率进行的新研究。模块化形式的P-ADIC斜率是其p-th Hecke特征值的频率被固定的prime p分组。自1980年代以来，对斜坡的预测和定理就已经存在。七年前，伯格达尔（Bergdall）和波拉克（Pollack）提出了一种几乎所有先前所有想法的方式（“幽灵猜想”）。幽灵猜想的输入是模块化形式的一致性类别。输出是同类斜率的基本配方。主要警告是Ghost猜想仅适用于“常规”类。但是，假设有规律性，Liu，Truong，Xiao和Zhao（LTXZ）最近建立了猜想。当前的项目删除了幽灵猜想中的常规假设。新工具是Galois表示形式的Emerton和Gee（EG）Moduli堆栈。用galois术语，规律性是EG堆栈中的通用属性。因此，该项目的技术创新是在堆栈上变形了斜率问题。几何重新制定将打开概括LTXZ证明的大门。它还将为希尔伯特模块化形式或更高排名的自身形式的新颖研究创造空间。该奖项反映了NSF的法定任务，并认为使用基金会的知识分子优点和更广泛的影响评估标准，认为值得通过评估来支持。