Galois Structures and Arithmetic Statistics

伽罗瓦结构和算术统计

• 批准号: 2200541
• 负责人: Yuan Liu
• 金额: $ 14.14万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-15 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200541&HistoricalAwards=false
• 关键词: Galois; Structures; Arithmetic; Statistics

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.

代数数域的研究，即从具有整数系数的多项式的解中获得的数学对象，是数论中非常活跃的研究领域。与每个代数数域相关联，都有一个不变量，即类群，用于衡量数域结构的复杂性。虽然类群的计算一般来说很困难，但数值证据表明，当在合理的数域族范围内时，类群是“随机的”，并且这种随机行为在最近活跃的领域——算术统计中进行了研究。研究这种随机不变量的一种方法是构建一个包含关键参数的随机模型，并使用该模型来模拟类组的分布。该项目将探讨有关此类随机模型的构建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。 该项目的研究是代数数论、算术统计、群论和概率论的交叉点。 PI 和她的合作者制作了一系列关于非阿贝尔 Cohen-Lenstra 启发式的作品，以给出合理的猜想来预测全局域的最大无分支扩展的伽罗瓦群的分布（这是类群的非阿贝尔概括）。特别是，本文构建的非阿贝尔随机群模型在连接伽罗瓦对象与其算术统计量方面发挥着重要作用。 PI 计划继续沿着这个方向开展工作，其中包括：深入研究无分支扩展的伽罗瓦结构，证明新算术统计启发式所隐含的伽罗瓦结构的性质，以及通过消除先前工作中的一些限制来研究随机群模型如何受到影响。明确地说，该项目有三个关键目标：1）在基域包含统一根的情况下修改非阿贝尔Cohen-Lenstra启发式； 2）研究基场的签名如何影响分布，3）概括格思的猜想。该奖项反映了 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。