Class Groups of Number Fields and Zeros of L-functions

数论是研究整数，更具体地说是质数的数学分支。素数定理（描述素数在正整数中的分布）在1896年由Hadamard和de la Vallee Poussin通过理解黎曼ζ函数的某些性质独立证明。黎曼ζ函数及其推广称为l函数，是数论中普遍存在但又神秘的函数。这些函数可以与大量的数学对象（包括狄利克雷字符、数字字段和椭圆曲线）相关联来定义。理解l函数零点的位置是所有数学中的一个中心问题。虽然我们目前不能证明黎曼假说，黎曼在1859年提出，有许多卓有成效的研究，以追求更好地理解零点的l -函数。特别是，l函数的零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，它始于高斯的工作，以及（广义）黎曼假设的一种特殊类型反例的可能性。这些假设的反例被称为朗多-西格尔零，目前不能排除它们的存在。更具体地说，这个项目将研究解析数论和代数数论的交叉问题。它将研究关于l函数族的Chebotarev密度定理的应用，l函数的零的垂直分布，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。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.