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

整数是最实用和最重要的数学对象之一。人类已经研究了几千年。数论的目的是理解整数所具有的模式。基本问题围绕乘法：在某些序列中，偶数（即可被2整除）的频率是多少？可被3整除？还是五个？19世纪世纪的研究人员引入对称性来揭示数字中隐藏的模式。在20世纪70年代，罗伯特·朗兰兹（Robert Langlands）对对称性进行了意义深远的研究。从那时起，这些问题就一直困扰着数论家。他们预测对称行为所看到的模式同样会从复数的微积分（“模形式”）中产生。出现在两个地方的模式是数学互惠的一个例子。该项目将完善朗兰兹的互惠预测。新的工具是对称作用的几何空间，由Emerton和Gee在过去的15年中构建。这些空间被认为是将互易性问题转化为几何问题。这个项目建立了这种信念的实例。它将连接整除模式从世界上的模形式几何定理埃默顿和吉的空间。该项目具有广泛的影响。计算数据将包含在广泛使用的L函数和模块化形式数据库中。该项目还开发了用于教学的计算工具。开放教育资源（OER）是公共领域的学习材料。它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论和抽象代数学习的OER。该项目以另外两种方式支持教育和外联。首先，数学圈将在公立学校运行。第二，将制定研究项目，以支持青年科学家数学计划。最后，该项目计划在数论两个研究研讨会。两者都旨在传播数论和互惠的新进展。更详细的目标是一个新的研究的p-adic斜率的模形式和伽罗瓦表示。模形式的p-adic斜率是它的第p个Hecke特征值被固定素数p整除的频率。七年前，Bergdall和Pollack提出了一种方法（“幽灵猜想”）来统一几乎所有先前的想法。幽灵猜想的输入是模形式的同余类。输出是类中斜率的基本配方。主要的警告是幽灵猜想只适用于“常规”类。但是，假设规律性，刘，张，肖，赵（LTXZ）最近建立了猜想。目前的项目删除了幽灵猜想中的正则假设。新的工具是Emerton和Gee的Galois表示的模栈。在伽罗瓦术语中，正则性是EG栈上的一般属性。该项目的技术创新，从而变形斜坡问题的堆栈。一个几何重新制定将打开大门，推广LTXZ证明。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。

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.