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: 2302285
• 负责人: Robert Pollack
• 金额: $ 19.5万
• 依托单位: Trustees of Boston University
• 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=2302285&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.

整数是最实用和最重要的数学对象之一。人类对它们的研究已有数千年之久。数论旨在理解整数所具有的模式。基本问题围绕乘法：某个序列中的数字出现偶数的频率有多少（即能被二整除）？能被三整除吗？还是五个？十九世纪的研究人员引入对称作用来揭示数字中隐藏的模式。 20 世纪 70 年代，罗伯特·朗兰兹 (Robert Langlands) 对对称性做出了影响深远的猜想。从那时起，这些猜想就一直困扰着数论学家。他们预测对称作用所看到的模式将同样从复数微积分（“模形式”）中产生。出现在两个地方的模式是数学互易的一个例子。该项目将完善朗兰兹互易预测。新工具是对称作用的几何空间，由艾默顿和吉在过去十五年中构建。这些空间被认为可以将互易问题转化为几何问题。该项目确立了这一信念的实例。它将把模形式世界的可分性模式与艾默顿和吉空间的几何定理联系起来。该项目具有更广泛的影响。计算数据将包含在广泛使用的 L 函数和模块化表格数据库中。该项目还开发用于教学的计算工具。开放教育资源 (OER) 是放置在公共领域的学习材料。他们的主要好处是以低成本提供学习体验。它们可以进行调整以适应多样化的学习环境。该项目开发开放教育资源，用于基于计算机的数论和抽象代数学习。该项目以另外两种方式支持教育和推广。首先，数学圈将在公立学校开展。其次，将开发研究项目来支持青年科学家数学计划。最后，该项目计划举办两个数论研究研讨会。两者都旨在传播数论和互易性的新进展。更详细的目标是对模形式和伽罗瓦表示的 p 进斜率进行新的研究。模形式的 p 进斜率是指其第 p 个 Hecke 特征值被固定素数 p 整除的频率。关于斜率的预测和定理自 20 世纪 80 年代以来就已存在。七年前，伯格达尔和波拉克提出了一种统一几乎所有先前想法的方法（“幽灵猜想”）。幽灵猜想的输入是模形式的同余类。输出是该类斜坡的基本配方。主要警告是幽灵猜想仅适用于“常规”类。但是，假设有规律性，Liu、Truong、Xiao 和 Zhu (LTXZ) 最近提出了这个猜想。当前的项目消除了幽灵猜想中的常规假设。新工具是 Emerton 和 Gee (EG) 的伽罗瓦表示模栈。用伽罗瓦术语来说，正则性是 EG 堆栈上的通用属性。因此，该项目的技术创新正在改变堆栈上的斜率问题。几何重构将为 LTXZ 证明的推广打开大门。它还将为希尔伯特模形式或更高阶自同构形式的新颖研究创造空间。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。