Arithmetic Statistics, Fourier Analysis, and Equidistribution

数论是研究整数和整数的比率（称为有理数）。特别是，自古以来，数论家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，正方形能比立方体正好多22吗？该项目主要研究各种类型的随机方程具有整数或有理数解的概率。例如，在前一个授权期间，最近证明的一个此类结果是，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。目前的项目的目标是开发技术，以证明数论中其他类型的经典方程的类似结果。该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，其中包括表示论和分析的基本成分。在此奖项期间，该计划预计将导致各种类型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性的定量形式;关于一般群的塞尔默群分布的新结果（不一定是奇数次）超椭圆曲线，和新的结果的行为合理和积分点的曲线和高维品种对应的表示有一个环的不变量，不一定是自由的。这将涉及到从群论，表示论，几何的数字，傅立叶分析，和更多的技术相结合，产生一个令人兴奋的技术的相互作用，我们希望将有应用超出只是提到的问题。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.