Explicit Galois Deformation Theory, Modular Forms, and Iwasawa Theory

任何对分类的科学努力都要分两步进行：第一，对最基本的物体进行分类，第二，对基本物体可以组合成更复杂物体的不同方式进行分类。例如，在化学中的分子分类中，最基本的对象是出现在元素周期表上的化学元素，而组合是它们之间可以形成的各种键。在代数数论中，这个项目的主题是对称为伽罗瓦表示的代数结构进行分类，这种代数结构捕捉到“丢番图方程”的解的对称性，多项式方程是我们在整数中寻找解的地方。基本对象称为“不可还原的”，它们的组合方式称为“扩展”。即使在相对简单的不可约部分之间，也有非常丰富的扩张理论。在这个项目中，PI的目标是建立猜想的实例，这些猜想通过来自其他数学领域的对象来参数化这些扩展的结构，包括分析和几何。指导原则来自形变理论：从两个不可约对象之间最简单的可能扩张开始，如果有许多方法使其变形，那么肯定还有许多其他扩张，而如果它更刚性，扩张的结构就更简单。当可以找到显式变形时，这将提供有关扩展的信息。形变理论在现代数论中起着至关重要的作用，在Andrew Wiles的《费马大定理的证明》一书中起着核心作用。PI将通过系统地使用伪表示来发展研究可约表示的形变理论的技术。在这种情况下，显式变形来自于Eisenstein级数和尖顶模形式之间的同余。通过对形变的分析，得到了与布洛赫-加藤猜想所预言的L函数的特定值有关的塞尔默群大小的结果。特别令人感兴趣的是精细化的猜想，如Sharifi猜想，它从同余的更微妙的性质中提取关于Selmer群的更精细的信息。这一奖项反映了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。

Any scientific effort at classification proceeds in two steps: first, classify the most basic objects, and, second, classify the different ways the basic objects can be combined to form more complex objects. For example, in the classifications of molecules in chemistry, the most basic objects are the chemical elements appearing on the periodic table, and the combinations are the various bonds that can form between them. In algebraic number theory, the subject of this project, the goal is to classify algebraic structures called Galois representations, which capture the symmetries of solutions to "Diophantine equations," polynomial equations where we seek solutions among the integers. The basic objects are called "irreducible," and the ways they can be combined are called "extensions." There is a very rich theory of extensions even between relatively simple irreducible pieces. In this project, the PI aims to establish instances of conjectures that parameterize the structure of these extensions by objects coming from other areas of mathematics, including analysis and geometry. The guiding principle comes from deformation theory: starting with the simplest possible extension between two irreducibles, if there are many ways to deform it, then there must be many other extensions, whereas if it is more rigid, the structure of extensions is simpler. When explicit deformations can be found, this gives information about extensions. Deformation theory is crucial in modern number theory and played a central role in Andrew Wiles's proof of Fermat's Last Theorem.In this project, the extension groups to be analyzed are Selmer groups. The PI will develop techniques for studying the deformation theory of reducible representations by systematically using pseudorepresentations. In this case, explicit deformations come from congruences between Eisenstein series and cuspidal modular forms. Analysis of deformations leads to results relating the size of Selmer groups to special values of L-functions, as predicted by the Bloch-Kato conjectures. Of particular interest are refined conjectures, such as Sharifi's conjecture, which extract finer information about Selmer groups from more delicate properties of congruences.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.