Explicit Methods for Finding Rational Points on Varieties

Ohio State University, The, Columbus OH

Investigators

Abstract

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.