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。人类已经研究它们几千年了。数论旨在理解整数所具有的模式。基本问题围绕乘法展开：在某些数列中数字偶数的频率是多少（即能被2整除）？能被3整除吗？还是5个?19世纪的研究人员引入对称行为来揭示数字中隐藏的模式。在20世纪70年代，罗伯特·朗兰兹对对称性做出了深远的猜想。从那时起，这些猜想就一直占据着数论学家的脑海。他们预测对称行为所看到的模式同样会从复数演算（“模形式”）中产生。出现在两个地方的模式是数学互易性的一个例子。该项目将完善朗兰兹的互易性预测。新的工具是对称作用的几何空间，是由Emerton和Gee在过去15年里构建的。这些空间被认为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。它将把模形式世界中的可分模式与emton和Gee空间中的几何定理联系起来。该项目具有实质性的广泛影响。计算数据将包括在广泛使用的l函数和模块形式数据库中。该项目还开发了用于教学的计算工具。开放教育资源（OERs）是指放置在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。它们可以适应不同的学习环境。该项目为基于计算机的数论和抽象代数学习开发OERs。该项目还通过另外两种方式支持教育和外联活动。首先，数学圈将在公立学校开展。第二，将发展研究项目，以支持青年科学家数学计划。最后，本计划计划两个数论研究工作坊。两者都旨在传播数论和互易性方面的新进展。更详细的目标是模形式和伽罗瓦表示的p进斜率的新研究。模形式的p进斜率是它的第p个赫克特征值被一个固定素数p整除的频率。自20世纪80年代以来，关于斜率的预测和定理就已经出现了。七年前，伯格道尔和波拉克提出了一种方法（“幽灵猜想”）来统一几乎所有先前的想法。鬼猜想的输入是模形式的同余类。输出是类中斜率的基本公式。主要的警告是，幽灵猜想只适用于“常规”类。但是，Liu, Truong, Xiao, and Zhao （LTXZ）最近建立了这个猜想。当前的项目去掉了鬼猜想中的常规假设。新的工具是Emerton和Gee的伽罗瓦表示的模堆栈。在伽罗瓦术语中，正则性是EG堆栈上的一般性质。因此，该项目的技术创新正在改变堆栈上的坡度问题。几何形式的重新表述将为LTXZ证明的推广打开大门。它也将为希尔伯特模形式或更高阶自同构形式的新研究创造空间。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.