CAREER: Exceptional Points on Modular Curves

该奖项全部或部分根据2021年美国救援计划法案（公法117-2）资助。椭圆曲线是现代数论中最普遍的对象之一。它们在理论数学（例如费马大定理的证明）和信息安全方面都有深远的应用，它们构成了通常用于提供安全Web浏览的密码系统的基础。该项目的研究重点是椭圆曲线，通过将这些曲线视为称为模曲线的几何对象上的区别点，揭示了意想不到的算术特性。在这种情况下，该项目将开发新的工具来识别这些不寻常的椭圆曲线，利用模曲线的几何形状和相关的代数结构。此外，该项目还包括几个教育部分，例如一个培训计划，其中硕士生将担任研究探索课程本科生的项目负责人。该项目的一个中心目标是扩大参与数学科学，无论是在本科生和研究生水平，这项研究的主要目标是解释孤立或零星的点上的模曲线，特别是在这种情况下，这些点对应于椭圆曲线的点（或有理循环等距）的高阶定义在一个数域的异常低的程度。这是出于一个愿望，以控制存在这样的点在无限家庭的模块化曲线，这是在心脏的开放问题所提出的马祖尔和塞尔。一个工具的组合，将采用，包括几何方法源于Arakelov相交理论和显式计算技术有关的伽罗瓦表示的椭圆曲线。对于特定的模数曲线，该项目追求的是与具有复数乘法的椭圆曲线相对应的孤立点与那些无法用任何已知的几何或模数现象解释的孤立点之间的类比。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). Elliptic curves are among the most ubiquitous objects in modern number theory. They have far-reaching applications, both in theoretical mathematics – such as in the proof of Fermat's Last Theorem – and in information security where they form the basis of a cryptosystem commonly used to provide secure web browsing. The research in this project focuses on elliptic curves with unexpected arithmetic properties revealed by viewing these curves as distinguished points on a geometric object called a modular curve. In this context, the project will develop new tools for identifying these unusual elliptic curves, exploiting both the geometry of the modular curve and associated algebraic structures. In addition, the project includes several educational components, such as a training program in which master's degree students will serve as project leaders for undergraduates enrolled in a research exploration course. A central aim of the project is to broaden participation in the mathematical sciences, both at the undergraduate and graduate level.The main goal of this research is to explain isolated or sporadic points on modular curves, especially in the case where such points correspond to elliptic curves with a point (or a rational cyclic isogeny) of high order defined over a number field of unusually low degree. This is motivated by a desire to control the existence of such points in infinite families of modular curves, which lies at the heart of open questions raised by Mazur and Serre. A combination of tools will be employed, including geometric approaches stemming from Arakelov intersection theory and explicit computational techniques relating to Galois representations of elliptic curves. For certain modular curves, the project pursues an analogy between isolated points corresponding to elliptic curves with complex multiplication and those whose existence fails to be explained by any known geometric or modular phenomenon.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.