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

确定多项式方程的整数解是一个活跃的研究领域，至少有两千年的历史。 然而，许多问题仍然存在，这些方程仍然是至关重要的，因为用于研究它们的技术帮助塑造了现代密码系统的基础。在1922年，Louis Mordell证明了定义亏格至少为2的曲线的方程只有2个有理解。 Gerd Faltings在1983年证明了这一点，但他的证明并没有明确地给出这些曲线上的有理点集。 数学上确定这个集合是数论中最基本的开放问题之一。 二次Chabauty是一种确定有理点集的新方法，通过理论和计算策略的结合，PI将给出二次Chabauty算法来确定新类曲线上的有理点。该项目还包括几个教育和外联部分，包括在关岛举办一系列以本科生为重点的关于计算工具的讲习班，旨在扩大传统上代表性不足的群体对科学、技术、工程和数学的参与。PI还将为波士顿地区的高中生共同组织为期一周的数学探索和计算暑期课程，以及数学科学研究所的丢番图几何学期课程。主要研究主题集中在使用p-adic高度确定亏格至少为2的曲线上的有理点的算法。它们包括以下内容：使用p-adic高度产生模曲线的二次Chabauty算法，开发椭圆二次Chabauty算法来研究扭曲的Fermat曲线，并在族中建立科尔曼积分和p-adic高度的基础设施。这些新算法将在大型曲线数据库上运行，所产生的数据将被分析并与数学界共享。这个奖项反映了NSF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。

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.