Arithmetic Statistics, Fourier Analysis, and Equidistribution

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，在前一次资助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，不存在x和y的任何有理数解。本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。这项拟议的活动将涉及大量研究生、本科生和博士后，他们将接受最新技术的培训，以帮助解决推动该领域发展的基本问题。该项目是正在进行的研究计划的一部分，旨在解决数论和算术几何中的基本问题，基本成分来自表示理论和分析。在该奖项期间，该计划有望导致关于各类代数对象的伽罗瓦群分布的进一步结果，从而产生希尔伯特不可约性的新的有效的定量形式；关于一般(不一定是奇数次)超椭圆曲线的塞尔默群分布的新结果；以及关于曲线和高维变种上的有理和积分点的行为的新结果，对应于具有不一定自由的不变量环的表示。这将涉及到结合群论、表示论、数字几何、傅立叶分析等技术，产生令人兴奋的相互作用的技术，我们预计这些技术将具有超出上述问题的应用。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. The proposed activity would heavily involve a number of graduate students, undergraduates, and postdocs, who would be trained in the latest techniques in order to help address basic questions that advance the field.This project is part of an ongoing research program addressing fundamental questions in number theory and arithmetic geometry, with essential ingredients from representation theory and analysis. During the period of this award, this program is expected to lead to further results on the distribution of Galois groups of various types of algebraic objects, thus yielding new effective, quantitative forms of Hilbert irreducibility; new results on the distribution of Selmer groups of general (not necessary odd degree) hyperelliptic curves, and new results on the behavior of rational and integral points on curves and higher-dimensional varieties corresponding to representations having a ring of invariants that is not necessarily free. This will involve combining techniques from group theory, representation theory, geometry of numbers, Fourier analysis, and more, yielding an exciting interplay of techniques that we expect will have applications beyond just the problems mentioned.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.