LEAPS-MPS: Isolated Points on Curves

LEAPS-MPS：曲线上的孤立点

• 批准号: 2137659
• 负责人: Abbey Bourdon
• 金额: $ 18.94万
• 依托单位: Wake Forest University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2137659&HistoricalAwards=false
• 关键词: LEAPS; MPS; Isolated; Points; Curves

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.

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)。几千年来，多项式方程组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和计算机科学到数学生物学的各个领域得到了应用。理解这种方程组的解集的一种方法可以追溯到希腊数学家丢番图斯，就是试图只找到那些坐标在整数或有理数中的解。在我们的多项式方程定义了一条代数曲线的情况下，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合。这些可以被认为是方程组的“意想不到的”解，而这一提议的最终目标是开发新的工具来识别这些解何时发生。调查的几个部分适合研究生和其他职业生涯早期的研究人员，他们的参与构成了拟议工作的教育影响之一。第二个教育努力是一个研究性培训计划，包括即将到来的硕士学位学生和早期的本科生。在各级，该计划将积极招收传统上在科学界代表性较低的群体的学生。这一建议的中心目的是刻画亏格至少为2的模曲线上的孤立点，其动机是与几个著名的分类问题和开放猜想有关。对于固定模数曲线族，所提出的工作主要分为两大类：(1)求出某一类椭圆曲线对应的所有任意次孤立点；(2)求出任意椭圆曲线对应的所有固定次数孤立点。该项目寻求对孤立点的几种新解释，要么是通过利用与曲线的雅可比相关的代数结构，要么是通过阿拉克洛夫交集理论。一个具体的目的是应用这些结果来解释二次域上椭圆曲线的某些意想不到的同构。在不同的方向上，PI将研究与Q曲线相关的孤立点的类别，其动机与Serre关于附加在椭圆曲线上的伽罗瓦表示的一致性猜想有关。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。