Concrete Arithmetic Applications of Analytic Number Theory

解析数论的具体算术应用

• 批准号: 1601398
• 负责人: Robert Lemke Oliver
• 金额: $ 12.74万
• 依托单位: Tufts University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-15 至 2020-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1601398&HistoricalAwards=false
• 关键词: Concrete; Arithmetic; Applications; Analytic; Number

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.

数论是研究整数的数学的分支。 在这个研究项目中考虑的两个基本对象是素数和多项式方程的整数解，这两者都在密码学中发挥着核心作用。 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的结构。 在数论中，这些族通常是离散的--没有办法在不离开整数的情况下连续地将一个整数变成另一个整数--但是人们可以采取广泛的观点，询问满足某个性质的问题或对象的比例;这是数论的子领域算术统计的观点。 本项目将从不同的角度探讨算术统计和数论中的各种问题，目的是了解这些不同方法的互补程度和相互信息。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。 这个项目将研究增长的平均规模的塞尔默组的椭圆曲线水平结构使用的工具，从几何的数字作为开发的Bhargava。 它将考虑二次扭曲的椭圆曲线，特别是在何种程度上分析和代数非零的结果可以被证明，它将考虑类似的问题扭曲的椭圆曲线的nonabel数领域。它还将研究Artin L-函数的值的分布，获得类数，极值问题和几何的应用。