Explicit Galois Deformation Theory, Modular Forms, and Iwasawa Theory

任何科学的分类工作都是分两步进行的：第一步，对最基本的对象进行分类;第二步，对基本对象组合成更复杂对象的不同方式进行分类。例如，在化学中对分子的分类中，最基本的对象是出现在周期表上的化学元素，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，目标是分类称为伽罗瓦表示的代数结构，它捕获了“丢番图方程”的解的对称性，多项式方程，我们在整数中寻找解。基本的对象被称为“不可约”，它们组合的方式被称为“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的目标是建立一些实例，这些实例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。指导原则来自变形理论：从两个不可约之间最简单的可能扩张开始，如果有很多方法可以使它变形，那么一定有很多其他扩张，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，这给出了关于延伸的信息。变形理论在现代数论中是至关重要的，在安德鲁·怀尔斯的费马大定理的证明中起了核心作用。在这个项目中，要分析的扩张群是塞尔默群。PI将通过系统地使用伪表示来开发研究可约表示的变形理论的技术。在这种情况下，显式变形来自爱森斯坦级数和尖点模形式之间的同余。变形分析导致的结果有关的大小塞尔默组的特殊值的L-功能，预测的布洛赫-加藤图解。特别令人感兴趣的是精炼的猜想，如Sharifi猜想，它从更微妙的同余性质中提取关于塞尔默群的更精细的信息。该奖项反映了NSF的法定使命，并被认为值得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来支持。

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.