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.

对代数数字字段的研究，即具有整数系数的多项式溶液获得的数学对象，是数量理论的一个非常活跃的研究领域。与每个代数数字字段相关联，有一个不变的类别组，可以测量数字字段结构的复杂性。尽管总体上很难计算，但数值证据表明，在合理的数字领域家庭范围内，班级是“随机的”，并且在最近活跃的领域 - 算术统计中研究了这种随机性行为。研究这种随机不变的一种方法是构建一个随机模型，该模型包含关键参数并使用该模型模拟类组的分布。该项目将探讨有关此类随机模型及其应用的构造的各种问题。该项目还为本科研究经验提供了支持。该项目的研究是在代数数理论，算术统计，群体理论和概率理论的交集。 PI和她的合作者在非亚洲Cohen-Lenstra启发式方法上制作了一系列作品，以合理的猜想来预测Galois群体的分布，包括全球全球领域的最大未受到的不受影响的扩展（这是分支组的非 - 阿伯岛概括）。特别是，在这项工作中构建的非亚伯随机组模型在将Galois对象与其算术统计数据连接起来起着重要作用。 PI计划继续沿着这个方向进行工作，其中包括：深入研究未受到扩展的Galois结构，提供了Galois结构的特性，这些属性是由新的算术统计启发式所暗示的，并研究了如何通过去除先前工作的某些限制来影响随机组模型。明确地，该项目具有三个关键目标：1）在基本场含有统一根源的情况下修改非亚伯群岛的启发式方法； 2）研究基本场的签名如何影响分布，3）概括Gerth的概念。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响评估标准来评估，被视为珍贵的支持。