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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.

数论是数学的分支，主要研究整数，更具体地说，研究素数。素数定理（描述了素数在正整数中的分布）在1896年由阿达玛和德·拉·瓦利·普桑通过理解黎曼ζ函数的某些性质独立证明。黎曼zeta函数及其推广，称为L函数，是数论中普遍存在但神秘的函数。这些函数可以与大量的数学对象相关联地定义，包括狄利克雷字符、数域和椭圆曲线。理解L-函数零点的位置是所有数学中的中心问题。虽然我们目前无法证明黎曼假设，提出了黎曼在1859年，有许多富有成效的调查，以追求更好地了解零点的L-函数。特别是，L-函数零点的垂直分布与另外两个中心问题有着深刻的联系：类数问题，它起源于高斯的工作，以及（广义）黎曼假设的特殊类型反例的可能性。这些假设的反例被称为朗道-西格尔零点，目前还不能排除它们的存在。更具体地说，这个项目将追求分析和代数数论的交叉问题。它将研究与Chebotarev密度定理有关的应用，用于L-函数族，L-函数零点的垂直分布，以及数域的类数。该项目的活动也将在指导研究生和本科生在文科设置方面产生更广泛的影响。该奖项反映了NSF的法定使命，并已被认为是值得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的支持。