Arithmetic Statistics, Fourier Analysis, and Equidistribution

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。拟议中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方面的培训，以帮助解决推动该领域发展的基本问题。该项目是一个正在进行的研究项目的一部分，该项目涉及数论和算术几何的基本问题，其中包含了表示理论和分析的基本成分。在此奖项期间，该计划预计将导致关于各种代数对象的伽罗瓦群分布的进一步结果，从而产生新的有效的，希尔伯特不可约的定量形式；关于一般（不一定是奇次）超椭圆曲线的Selmer群的分布的新结果，以及曲线和高维变量上的有理点和积分点的行为的新结果，这些变化对应于具有不一定是自由的不变量环的表示。这将涉及到群论、表示理论、数字几何、傅立叶分析等技术的结合，产生令人兴奋的技术相互作用，我们期望这些技术的应用将超出所提到的问题。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. 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.