CAREER: Exceptional Points on Modular Curves

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。椭圆曲线是现代数论中最普遍的对象之一。它们在理论数学中有着深远的应用，例如在费马大定理的证明中，在信息安全方面，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。这个项目的研究集中在具有意想不到的算术性质的椭圆曲线上，通过将这些曲线视为称为模曲线的几何对象上的区别点来揭示这些曲线。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，利用模曲线的几何和相关的代数结构。此外，该项目包括几个教育组成部分，例如一个培训计划，在该计划中，硕士研究生将担任参加研究探索课程的本科生的项目负责人。该项目的一个中心目标是扩大对数学科学的参与，包括本科生和研究生。本研究的主要目标是解释模曲线上的孤立或零星的点，特别是当这些点对应于椭圆曲线的情况下，这些点在异常低次的数域上定义了一个高阶的点(或有理循环同源)。这是出于希望控制模曲线无穷族中这类点的存在，这是Mazur和Serre提出的公开问题的核心。将使用多种工具，包括源于阿拉克洛夫交集理论的几何方法和与椭圆曲线的伽罗瓦表示有关的显式计算技术。对于某些模曲线，该项目寻求与具有复数乘法的椭圆曲线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其存在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优势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.