Rational points on modular curves, and the geometry of arithmetic statistics

该项目将探索数论和代数几何中的各种主题。这些都是古老的研究领域，根植于解决多项式方程的基本问题，并受到具体应用的推动。例如，希腊天文学家阿波罗尼乌斯（公元前240-190年）发展了他的圆锥和椭圆理论，以促进天文学的研究。关于数字和形状的问题仍然是数学研究的前沿，这个项目特别强调使用现代技术工具来研究经典问题。该项目包括本科生和研究生可以接触到的问题，并包括大量以学生为中心的会议组织（如亚利桑那冬季学校）。Mazur的扭转定理和等根定理是算术几何的基石，而算术统计是一个充满经典问题的古老领域。近年来，这两个领域都有了大量的新思想和新进展，特别是通过数字几何、模空间、代数拓扑、计算数论等方面的思想。特别是，该项目将研究Mazur的“程序B”，椭圆曲线上的高阶扭转，Batyrev- Manin和Malle猜想的堆栈推广（在某种意义上，这些猜想的插值），以及非阿贝尔（和无限次）Cohen- Lenstra启发（以及，在函数域情况下，定理）。这些子项目中的每一个都将引入新的方法和工具包/框架，这些方法和工具包/框架预计将广泛有用，并提出许多开放的问题和新的研究方向。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。

The project will explore various topics within number theory and algebraic geometry. These are ancient areas of inquiry rooted in very basic questions about solving polynomial equations and motivated by concrete applications. For example, the Greek astronomer Apollonius of Perga (240-190BC) developed his theory of conics and ellipses to facilitate the study of Astronomy. Questions about numbers and shapes still remain central to the frontier of mathematical research, and this project has a particular emphasis on using modern technical tools to study classical problems. The project includes problems accessible to undergraduates and graduate students, and includes efforts including substantial student focused conference organization (such as the Arizona Winter School).Mazur's torsion and isogeny theorems are cornerstones of arithmetic geometry, and arithmetic statistics is an old field full of classical problems. In recent years both areas have enjoyed an influx of new ideas and progress, especially via ideas from the geometry of numbers, moduli spaces, algebraic topology, computational number theory, and more. In particular, this project will study Mazur's ``Program B'', higher degree torsion on elliptic curves, a generalization of the Batyrev--Manin and Malle conjectures to stacks (in a sense, an interpolation of these conjectures), and non-abelian (and infinite degree) Cohen--Lenstra heuristics (and, in the function field case, theorems). Each of these sub-projects will introduced new methods and toolkits/frameworks that are expected to be broadly useful, and suggests numerous open problems and new directions for research.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.