Number Theory, Representation Theory, and Arithmetic Geometry

数论研究的是整数和整数的比率（称为有理数）。特别是，数论学家自古以来就对寻找方程的整数和有理数解很感兴趣，比如y^2 = x^3 + 22——也就是说，正方形是否正好比立方体大22 ？本课题主要研究各种类型的随机方程具有整数解或有理数解的概率。例如，在之前的资助期间证明的这类最近的结果是，大多数形式为y^2 = a x^4 + b x^3 + c x^2 + d x + e的方程，其中a，b,c,d，e是整数，不具有x和y的任何有理数解。当前项目的目标是开发技术来证明数论中感兴趣的其他类型的经典方程的类似结果。该项目是一个正在进行的研究项目的一部分，旨在解决数论和算术几何的基本问题，其基本成分来自表示理论和数字几何。拟议的活动将大量涉及研究生、本科生和博士后，他们将接受最新技术的培训，以帮助提高知识水平。一些目标将是使用最近发展的新技术来理解已经研究了几千年的方程的整数解的分布，并将一些基本应用于代数数论中的分布问题。一个关键因素将是基于数字几何的筛法技术的适当发展。所提出的程序预计将导致的结果，例如：积分解的分布经典方程，如Thue方程和莫德尔方程；正比例的立方场不是单基因的，尽管局部是单基因的；至少80%的椭圆曲线满足BSD猜想；不符合哈塞原理和积分哈塞原理的有理数的新变种族；以及对其他全局领域的概括。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 study of whole numbers, and ratios of whole numbers (called rational numbers). In particular, number theorists since antiquity have been interested in finding whole number and rational number solutions to equations, such as y^2 = x^3 + 22 - that is, can a square be exactly 22 more than a cube? This project largely concerns the study of the probability that random equations of various types have whole number or rational number solutions. For example, a recent result of this kind proven during the period of the previous grant is that most equations of the form y^2 = a x^4 + b x^3 + c x^2 + d x + e, where a,b,c,d,e are whole numbers, do not possess any rational number solutions for x and y. The goal of the current project is to develop techniques for proving similar results for other types of classical equations of interest in number theory.This project is part of an ongoing research program addressing fundamental questions in number theory and arithmetic geometry, with essential ingredients being employed from representation theory and the geometry of numbers. The proposed activity will heavily involve a number of graduate students, as well as undergraduate students and postdocs, who would be trained in the latest techniques in order to help advance the state of knowledge. Some of the goals would be to use recently developed as well as new techniques to understand the distribution of integer solutions to equations that have been studied for millennia, with some fundamental applications to distribution questions in algebraic number theory. A key ingredient will be a suitable development of sieve techniques based on the geometry of numbers. The proposed program is expected to lead to results on, e.g.: the distribution of integral solutions to classical equations such as the Thue equation and the Mordell equation; a positive proportion of cubic fields are not monogenic despite being locally so; at least 80% of all elliptic curves satisfy the BSD conjecture; new families of varieties over the rational numbers failing the Hasse principle and integral Hasse principle; and generalizations over other global fields.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.