Galois Structures and Arithmetic Statistics

代数数域是数论中一个非常活跃的研究领域，它是由整系数多项式的解得到的数学对象。与每个代数数域相关联的是一个不变量，即类组，用于度量数域结构的复杂性。虽然类群的计算通常很困难，但数值证据表明，当类群在一组合理的数域范围内时，类群是随机的，这种随机性行为是在最近活跃的一个领域-算术统计学中进行研究的。研究这种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，并用该模型来模拟班级群的分布。这个项目将探索关于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体验提供了支持。该项目的研究处于代数论、算术统计、群论和概率论的交叉点上。Pi和她的合作者已经在非阿贝尔Cohen-Lenstra启发式上产生了一系列工作，以给出合理的猜测来预测全局场(这是类群的非阿贝尔推广)的Galois群的分布。特别是，本文所构造的非阿贝尔随机群模型在连接伽罗瓦天体与其算术统计量方面发挥了重要作用。PI计划沿着这个方向继续工作，包括：深入研究非分支扩张的Galois结构，证明新的算术统计启发式所蕴含的Galois结构的性质，以及通过取消以前工作中的一些限制来研究随机群模型是如何受到影响的。明确地说，这个项目有三个关键目标：1)在基场包含单位根的情况下修改非阿贝尔的Cohen-Lenstra启发式；2)研究基场的签名如何影响分布，以及3)推广Gerth的猜想。这个奖项反映了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。

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.