CAREER: Exceptional Points on Modular Curves

该奖项全部或部分由《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。它们有着深远的应用，无论是在理论数学中——比如在费马大定理的证明中——还是在信息安全中，它们构成了通常用于提供安全网络浏览的密码系统的基础。该项目的研究重点是具有意想不到的算术性质的椭圆曲线，通过将这些曲线视为称为模曲线的几何对象上的不同点来揭示这些曲线。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，利用模块化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组成部分，例如一个培训计划，其中硕士学位的学生将担任参加研究探索课程的本科生的项目负责人。该项目的中心目标是扩大本科生和研究生对数学科学的参与。本研究的主要目的是解释模曲线上的孤立点或零星点，特别是在这种情况下，这些点对应于椭圆曲线上的高阶点（或有理循环等构），这些点定义在异常低次的数场上。这样做的动机是想要在无限的模曲线族中控制这些点的存在，这是Mazur和Serre提出的开放问题的核心。将使用多种工具，包括源自Arakelov交点理论的几何方法和与椭圆曲线伽罗瓦表示相关的显式计算技术。对于某些模曲线，该项目追求与具有复杂乘法的椭圆曲线对应的孤立点与无法用任何已知几何或模现象解释其存在的点之间的类比。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。

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.