Explicit Methods for Finding Rational Points on Varieties

寻找品种有理点的显式方法

• 批准号: 1902199
• 负责人: Jennifer Park
• 金额: $ 21.61万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1902199&HistoricalAwards=false
• 关键词: Explicit; Methods; Finding; Rational; Points

The project concerns various problems tied to number theory and algebraic geometry. Specifically, finding rational solutions to a system of polynomial equations has been the subject of active research since the ancient times. These rational points are quite sparse; Faltings (1983) proved that there are only finitely many rational points on curves of genus at least 2, and the minimalist conjecture asserts that most elliptic curves have low ranks as well. The sparsity of rational points have an application in the field of cryptography, as the explicit computation of various invariants of elliptic curves is often difficult.This project aims to further this point of view using various techniques coming from number theory, algebraic geometry, and logic. On one hand, the PI wishes to develop explicit p-adic methods that help find these rational points. On the other hand, heuristic arguments that approximate the difficulty of these methods can be helpful. These heuristics can range from finding a simple model for certain arithmetic invariants tied to the distribution of rational points, to arguments based in logic, to the estimate of computational complexity in the explicit computation of these invariants. These heuristics could have real-life consequences in, for example, approximating the hardness of isogeny-based cryptography.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.

该项目涉及与数论和代数几何有关的各种问题。具体地说，自古以来，寻找多项式方程组的有理解一直是活跃研究的主题。这些有理点是相当稀疏的；Faltings(1983)证明了亏格至少为2的曲线上只有有限多个有理点，极小主义猜想断言大多数椭圆曲线也有低阶。有理点的稀疏性在密码学领域有着广泛的应用，因为椭圆曲线的各种不变量的显式计算往往是困难的，本项目旨在利用数论、代数几何和逻辑的各种技术来进一步说明这一观点。一方面，PI希望开发出显式的p-adi方法来帮助找到这些有理点。另一方面，与这些方法的难度相近的启发式论证可能会有所帮助。这些启发式方法的范围可以从为与有理点的分布相关的某些算术不变量找到一个简单的模型，到基于逻辑的论点，再到在这些不变量的显式计算中的计算复杂性的估计。这些启发式算法可能会产生现实生活中的后果，例如，接近基于同源的密码学的难度。这一裁决反映了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。