Class Groups of Number Fields and Zeros of L-functions

L 函数的数域和零的类组

基本信息

批准号：
1902193
负责人：
Caroline Turnage-Butterbaugh
金额：
$ 7.5万
依托单位：
Carleton College
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1902193&HistoricalAwards=false
关键词：
Class Groups Number Fields Zeros

项目摘要

Number theory is the branch of mathematics concerned with studying the integers, and more specifically, the primes. The Prime Number Theorem (which describes the distribution of the primes among the positive integers) was proved in 1896 independently by Hadamard and de la Vallee Poussin by understanding certain properties of the Riemann zeta-function. The Riemann zeta-function and its generalizations, called L-functions, are ubiquitous yet mysterious functions in number theory. These functions can be defined in association with a plethora of mathematical objects, including Dirichlet characters, number fields, and elliptic curves. Understanding the location of the zeros of L-functions is a central problem in all of mathematics. While we cannot presently prove the Riemann Hypothesis, posed by Riemann in 1859, there are many fruitful investigations to pursue to better understand the zeros of L-functions. In particular, the vertical distribution of the zeros of L-functions has deep connections to two other central problems: the class number problem, which has its beginnings in the work of Gauss, and the possibility of a special type of counterexample to the (Generalized) Riemann Hypothesis. These hypothetical counterexamples are called Landau-Siegel zeros, and presently their existence cannot be ruled out.More specifically, this project will pursue problems in the intersection of analytic and algebraic number theory. It will study applications related to the Chebotarev density theorem for families of L-functions, the vertical distribution of zeros of L-functions, and class numbers of number fields. The activities of the project will also have broader impacts in terms of mentoring both graduate and undergraduate students in a liberal arts setting.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.

数字理论是与研究整数有关的数学分支，更具体地说是素数。素数定理（描述了正整数中素数的分布）在1896年通过Hadamard和de la Vallee Poussin独立证明，通过了解Riemann Zeta功能的某些特性。 Riemann Zeta功能及其概括（称为L功能）在数字理论中是无处不在但神秘的功能。这些函数可以与大量数学对象相关联，包括Dirichlet字符，数字字段和椭圆曲线。在所有数学中，了解L功能的零位置是一个核心问题。尽管我们目前无法证明Riemann在1859年提出的Riemann假设，但仍有许多富有成果的调查可以更好地了解L功能的零。特别是，L功能的零件的垂直分布与其他两个中心问题有着深厚的联系：班级编号问题，该问题在高斯的工作中具有起点，以及（广义）Riemann假设的特殊类型的反例。这些假设的反例称为Landau-Siegel Zeros，目前不能排除它们的存在。更具体地说，该项目将在分析和代数数理论的交集中提出问题。它将研究与Chebotarev密度定理有关L功能家族的应用，L功能的零零和数字字段数量的垂直分布。该项目的活动在指导文科艺术环境中的研究生和本科生方面也将产生更大的影响。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估评估标准的评估值得支持的。