CAREER: New Directions in p-adic Heights and Rational Points on Curves

职业生涯：p-adic 高度和曲线上有理点的新方向

• 批准号: 1945452
• 负责人: Jennifer Balakrishnan
• 金额: $ 48.77万
• 依托单位: Trustees of Boston University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-01-15 至 2024-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1945452&HistoricalAwards=false
• 关键词: CAREER; New; Directions; adic; Heights; CAREER; New; Directions; adic; Heights

Determining whole number solutions to polynomial equations has been an active area of study for at least two millennia. Nevertheless, many questions remain, and these equations continue to be crucially important, as the techniques used to study them have helped shape the foundation of modern cryptosystems. In 1922, Louis Mordell conjectured that equations defining curves of genus at least 2 have only finitely many rational solutions. Gerd Faltings proved this in 1983, but his proof does not explicitly yield the set of rational points on these curves. Algorithmically determining this set is one of the most fundamental open problems in number theory. Quadratic Chabauty is a new approach to determining the set of rational points, and through a combination of theoretical and computational strategies, the PI will give quadratic Chabauty algorithms to determine rational points on new classes of curves. This project also includes several educational and outreach components, including a collection of undergraduate-focused workshops in Guam on the topic of computational tools, aimed at broadening participation of traditionally underrepresented groups in STEM. The PI will also co-organize a week-long summer program in mathematical exploration and computation for high school students in the Boston area, as well as a semester program at Mathematical Sciences Research Institute on Diophantine geometry.The main research themes are centered on algorithms for determining rational points on curves of genus at least 2, using p-adic heights.They include the following: using p-adic heights to produce a quadratic Chabauty algorithm for modular curves, developing an elliptic quadratic Chabauty algorithm to study twisted Fermat curves, and building infrastructure in Coleman integration and p-adic heights in families. These new algorithms will be run on large databases of curves, and the resulting data will be analyzed and shared with the mathematical community. This has the potential to yield new insight into refined hypotheses under which theorems can be proved, as well as more precise conjectures to investigate.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.

确定多项式方程式的全数解决方案至少是两千年的活跃研究领域。 然而，仍然存在许多问题，这些方程式仍然至关重要，因为用于研究它们的技术有助于塑造了现代密码系统的基础。 1922年，路易·莫德尔（Louis Mordell）猜想定义属曲线的方程至少有2个有限的理性解决方案。 格丁斯（Gerd Faltings）在1983年证明了这一点，但他的证据并未明确地产生这些曲线的理性点。 从数字理论中，算法确定这一集是最根本的开放问题之一。 二次chabauty是确定合理点集的一种新方法，通过理论和计算策略的结合，PI将提供二次chabauty算法，以确定新曲线类别的合理点。该项目还包括几个教育和外展成分，包括关岛关于计算工具主题的一系列以本科生为中心的研讨会，旨在扩大传统上代表性不足的STEM的参与。 PI还将为波士顿地区的高中学生共同组织为期一周的数学探索和计算计划，以及在数学科学研究所的学期课程，主要研究主题的主要研究主题是在算法上的中心，用于确定量子的曲线，使用P-ad的曲目，包括P-aD的曲目，包括P-Adic的高度。用于模块化曲线的二次chabauty算法，开发一种椭圆形二次chabauty算法，以研究Coleman一体化和家庭中P-Adic高度的扭曲Fermat曲线以及建筑基础设施。这些新算法将在大型曲线数据库上运行，将与数学社区分析并共享所得数据。这有可能对可以证明定理的精制假设进行新的见解，并进行调查。该奖项反映了NSF的法定任务，并且认为值得通过基金会的知识分子优点和更广泛的影响审查标准通过评估来获得支持。