Galois Structures and Arithmetic Statistics

代数数域是由整系数多项式的解所得到的数学对象，它的研究是数论中一个非常活跃的研究领域。每个代数数域都有一个不变量，即类群，它度量数域结构的复杂性。虽然计算的类组是很难在一般情况下，数值证据表明，类组是“随机”时，范围在一个合理的家庭的数域，这种随机行为是研究在最近活跃的领域-算术统计。研究这一随机不变量的一种方法是构造一个包含关键参数的随机模型，并用该模型模拟类组的分布。本项目将探讨有关这种随机模型的构造及其应用的各种问题。该项目还为本科生的研究经验提供支持。在这个项目中的研究是在代数数论，算术统计，群论和概率论的交叉点。PI和她的合作者已经产生了一系列关于非交换的Cohen-Lenstra算法的工作，以给出合理的结构来预测全局域的最大非分歧扩张的伽罗瓦群的分布（这是类群的非交换推广）。特别是，在这项工作中构建的非阿贝尔随机群模型在连接伽罗瓦对象与它们的算术统计中发挥了重要作用。PI计划继续沿着这一方向开展工作，包括：深入研究非分歧扩张的Galois结构，证明新的算术统计算法所隐含的Galois结构的性质，以及研究去除以前工作中的一些限制对随机群模型的影响。说明一下，这个项目有三个主要目标：1）在基域包含单位根的情况下，修改非交换的Cohen-Lenstra算法; 2）研究基场的特征如何影响分布，和3）的方法这个奖项反映了NSF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进行评估，被认为值得支持的搜索.

The study of algebraic number fields, mathematical objects obtained from the solutions of polynomials with integer coefficients, is a very active area of research in number theory. Associated to each algebraic number field, there is an invariant, the class group, that measures the complexity of the number field structure. While computation of the class group is hard in general, numerical evidence suggests that the class group is "random" when ranging over a reasonable family of number fields, and this randomness behavior is studied in a recently active area – arithmetic statistics. One way to study this random invariant is to construct a random model which contains key parameters and use the model to simulate the distributions of class groups. This project will explore various questions about the constructions of such random models and their applications. The project also provides support for an undergraduate research experience. The research in this project is at the intersection of algebraic number theory, arithmetic statistics, group theory, and probability theory. The PI and her collaborators have produced a series of works on the non-abelian Cohen--Lenstra heuristics to give reasonable conjectures to predict the distributions of the Galois groups of the maximal unramified extensions of global fields (which are non-abelian generalizations of class groups). In particular, the non-abelian random group models constructed in this work play an important role in connecting the Galois objects with their arithmetic statistics. The PI plans to continue working along this direction, which includes: deeply studying the Galois structures of unramified extensions, proving properties of the Galois structure that are implied by the new arithmetical statistical heuristics, and studying how the random group models are affected by removing some restrictions in previous work. Explicitly, this project has three key goals: 1) modify the non-abelian Cohen—Lenstra heuristics in the case when the base field contains roots of unity; 2) study how the signature of the base field affects the distribution, and 3) generalize Gerth's conjecture.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.