Statistical Questions in Number Theory and Arithmetic Geometry

数论是一个蓬勃发展的数学研究领域，与计算机科学和数字安全有着密切的联系。这个项目的主要目标是通过定量的透镜来加深我们对某些普遍存在的数论对象（如椭圆曲线、幂残模素数和经典算术函数，如欧拉的pi函数）的理解。这项工作的一个新颖之处在于，解决这些问题的方法利用了我们对“整数解剖”的理解，它描述了一个数字分解成组成部分（质因数）的典型方式。本课程将探讨三个主要的研究课题：(1)详细研究算术函数的值分布，其中一个特别的问题是确定一个数有许多因数分解的精确条件。这里的“分解”可以是有序的，也可以是无序的，并且我们可能希望对这些部分施加某些附加条件。(2)幂残数的分布。(3)椭圆曲线扭转结构的统计分布。对于(1)和(3)，“解剖整数”的结果将起关键作用。对于(2)，PI计划用利用更高互易律的代数方法来补充解决这类问题的现有方法——这些方法大多是解析的，涉及特征和。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 a thriving area of mathematics research with deep connections to both computer science and digital security. The principal goal of this project is to deepen our understanding of certain ubiquitous number-theoretic objects (such as elliptic curves, power residues modulo prime numbers, and classical arithmetic functions such as Euler's phi-function), by studying them through a quantitative lens. One novel feature of this work is an approach to these problems that takes advantage of our understanding of the "anatomy of integers" describing the typical way a number breaks down into component parts (prime factors).Three main topics of investigation will be pursued: (1) A detailed study of the value-distribution of arithmetic functions, one particular problem being to pin down precise conditions under which a number has many factorizations. Here "factorizations" may be ordered or unordered, and we may wish to impose certain additional conditions on the parts. (2) The distribution of power residues. (3) The statistical distribution of torsion structures of elliptic curves. For (1) and (3), results from the "anatomy integers" will play a critical role. For (2), the PI plans to supplement the existing approaches to such problems - which have been mostly analytic, involving character sums - with algebraic approaches that make use of higher reciprocity laws.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.