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都有着深刻的联系。这个项目的主要目的是加深我们对某些无处不在的数论对象(如椭圆曲线、模素数的幂剩余，以及经典算术函数如欧拉的Phi函数)的理解，通过定量的透镜来研究它们。这项工作的一个新特点是一种解决这些问题的方法，它利用了我们对“整数解剖”的理解，它描述了一个数分解成组成部分(质因数)的典型方式。主要研究三个主题：(1)详细研究算术函数的值分布，一个特别的问题是确定一个数有许多因式分解的精确条件。在这里，“因式分解”可以是有序的，也可以是无序的，我们可能希望对部件施加某些额外的条件。(2)剩余电量的分布。(3)椭圆曲线扭转结构的统计分布。对于(1)和(3)，“解剖整数”的结果将起到关键作用。对于(2)，PI计划用代数方法来补充解决这类问题的现有方法-这些方法大多是分析性的，涉及特征和-利用更高的互惠法则。这一裁决反映了NSF的法定使命，并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，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.