LEAPS-MPS: Isolated Points on 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。几千年来，多项式方程系统的研究一直是数学的中心主题，今天，相关技术已经在从密码学、计算机科学到数学生物学的各个领域得到了应用。理解这样一个方程组的解集的一种方法——这种方法可以追溯到希腊数学家丢芬图——是试图只找到那些以整数或有理数为坐标的解。在多项式方程定义代数曲线的情况下，这些有理解通常位于一个更大的有限集合中，称为曲线的孤立点集。这些可以被认为是方程组的“意外”解，本提案的最终目标是开发新的工具来识别这些解何时出现。调查的几个组成部分适合研究生和其他早期职业研究人员，他们的参与构成了拟议工作的教育影响之一。第二项教育努力是一项研究训练计划，涉及即将入学的硕士生和早期本科生。在所有级别，该计划将积极招收属于传统上在科学领域代表性不足的群体的学生。这个提议的中心目标是描述至少2属的模曲线上的孤立点，动机是与几个众所周知的分类问题和开放猜想的联系。对于固定的模曲线族，所提出的工作主要分为两大类：(1)找到与某一类椭圆曲线对应的任意次的所有孤立点；(2)找到与任意椭圆曲线对应的任意次的所有孤立点。该项目通过利用与曲线雅可比矩阵相关的代数结构或通过Arakelov交集理论，对孤立点进行了几种新的解释。一个具体的目的是应用这些结果来解释二次场上椭圆曲线的某些意想不到的等同源性。在另一个方向上，PI将研究与q曲线相关的孤立点的类别，这些点与椭圆曲线上的伽罗瓦表示的Serre的均匀性猜想有关。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). The study of systems of polynomial equations has been a central theme in mathematics for thousands of years, and today related techniques have found applications in fields ranging from cryptography and computer science to mathematical biology. One approach to understanding the solution set of such a system of equations—which dates back to the Greek mathematician Diophantus—is to try to find only those solutions with coordinates in the integers or rational numbers. In cases where our polynomial equations define an algebraic curve, these rational solutions often sit inside a larger, finite set called the set of isolated points of the curve. These can be thought of as “unexpected” solutions to the system of equations, and the ultimate goal of this proposal is to develop new tools for identifying when these solutions occur. Several components of the investigation are suitable for graduate students and other early-career researchers, and their involvement constitutes one of the educational impacts of the proposed work. A second educational endeavor is a research training program involving both incoming Master’s degree students and early undergraduates. At all levels, this program will actively recruit students belonging to groups traditionally underrepresented in the sciences. The central aim of this proposal is to characterize isolated points on modular curves of genus at least 2, motivated by ties to several well-known classification problems and open conjectures. For a fixed family of modular curves, the work proposed falls into two main categories: (1) find all isolated points of any degree corresponding to a certain class of elliptic curves or (2) find all isolated points of a fixed degree corresponding to any elliptic curve. The project pursues several new explanations for isolated points, either by exploiting algebraic structures associated to the curve’s Jacobian or via Arakelov intersection theory. A specific aim is to apply these results to explain certain unexpected isogenies of elliptic curves over quadratic fields. In a different direction, the PI will study classes of isolated points associated to Q-curves motivated by ties to Serre’s Uniformity Conjecture for Galois representations attached to elliptic curves.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.