Number Theory, Representation Theory, and Arithmetic Geometry

数论是研究整数和整数比率(称为有理数)的学科。特别是，自古以来，数学家们一直对寻找方程的整数和有理数解感兴趣，比如y^2=x^3+22--也就是说，一个平方能恰好比一个立方体多22个吗？这个项目主要是研究各种类型的随机方程有整数解或有理数解的概率。例如，在上一次拨款期间证明的这类最新结果是，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，不具有任何关于x和y的有理数解。当前项目的目标是开发技术来证明数论中其他类型的感兴趣的经典方程的类似结果。本项目是一个正在进行的研究项目的一部分，该项目解决数论和算术几何中的基本问题，主要成分来自表示理论和数的几何。拟议的活动将大量涉及一些研究生以及本科生和博士后，他们将接受最新技术的培训，以帮助提高知识水平。其中一些目标将是使用最近开发的以及新的技术来了解已被研究了数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分技术的适当发展。拟议的计划预计将导致以下方面的结果：经典方程(如Thue方程和Mordell方程)的积分解的分布；正比例的三次场尽管是局部的，但不是单源的；至少80%的所有椭圆曲线满足BSD猜想；不符合哈斯原理和积分哈斯原理的有理数上的新变种；以及对其他全球领域的概括。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.