Concrete Arithmetic Applications of Analytic Number Theory

数论是研究整数的数学分支。本研究项目中考虑的两个基本对象是素数和具有整数解的多项式方程，这两者在密码学中都发挥着核心作用。现代数学的一个主题是考虑一系列相关的问题，并了解解决方案如何随着问题在家庭中的变化而变化，着眼于揭示任何单个问题背后的结构。在数论中，族通常是离散的--在不离开整数的情况下，不可能连续地将一个整数变成另一个--但一个人可以从一个宽泛的角度，询问满足某种性质的问题或对象的比例；这是数论中被称为算术统计的子领域的观点。这个项目将通常从不同的角度探索算术统计和数论中的各种问题，目的是看看这些不同的方法相互补充和相互启发的程度。更具体地说，考虑的主要对象是素数、L函数、数域和椭圆曲线。这个项目将使用Bhargava发展的数几何中的工具来研究具有水平结构的椭圆曲线的Selmer群的平均尺寸的增长。它将考虑椭圆曲线的二次扭转，特别是解析和代数非零化结果可以相交的程度，并将考虑非阿贝尔数域的椭圆曲线扭转的类似问题。它还将研究Artin L函数的值分布，获得在类数、极值问题和几何中的应用。

Number theory is the branch of mathematics concerned with the study of the integers. Two of the fundamental objects considered in this research project are prime numbers and polynomial equations with integer solutions, both of which play a central role in cryptography. A theme of modern mathematics is to consider a family of related questions and to understand how the solution changes as the question varies within the family, with an eye toward revealing the structure underlying any individual question. In number theory, the families are typically discrete -- there is no way to continuously turn one integer into another without leaving the integers -- but one may take a broad view and ask about the proportion of questions or objects satisfying a certain property; this is the viewpoint of the subfield of number theory known as arithmetic statistics. This project will explore various questions within arithmetic statistics and number theory in general, often from different angles, with the goal being to see the extent to which these different approaches complement and inform each other.More specifically, the key objects of consideration are primes, L-functions, number fields, and elliptic curves. This project will study the growth of the average size of Selmer groups of elliptic curves with level structure using tools from the geometry of numbers as developed by Bhargava. It will consider quadratic twists of elliptic curves, in particular the extent to which analytic and algebraic non-vanishing results can be intersected, and it will consider similar problems for twists of elliptic curves by nonabelian number fields. It will also study the distribution of values of Artin L-functions, obtaining applications to class numbers, extreme value problems, and geometry.