Explicit Methods for Finding Rational Points on Varieties

该项目涉及与数论和代数几何有关的各种问题。具体地说，寻找多项式方程组的合理解自古以来一直是积极研究的主题。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1000个有理点，而最小猜想断言大多数椭圆曲线的秩也很低。有理点的稀疏性在密码学领域有应用，因为椭圆曲线的各种不变量的显式计算通常是困难的。本项目旨在使用来自数论，代数几何和逻辑的各种技术来进一步推进这一观点。一方面，PI希望开发显式p-adic方法来帮助找到这些合理的点。另一方面，近似这些方法的难度的启发式参数可能会有所帮助。这些算法的范围可以从找到一个简单的模型，为某些算术不变量绑在合理的点的分布，以论据为基础的逻辑，估计计算复杂性的显式计算这些不变量。这些密码学可能会对现实生活产生影响，例如，接近基于同源性的密码学的难度。该奖项反映了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。

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.