Rational Points on Modular Curves and Uniformity Questions

Brown University, Providence RI

Investigators

Abstract

This award concerns research in number theory. The study of polynomial equations has been a central theme in mathematics since ancient times. A vast amount of knowledge has been developed in pursuit of the solutions to specific polynomial equations. For example, in 1637 Fermat famously asked which nth powers are the sum of two other nth powers . Fermat's problem was resolved in the 1990s; the solution advanced insight into elliptic curves, which played a key role in the proof. Elliptic curves are central objects in cryptography, number theory, and algebraic geometry, and are an active area of research. A major open question is whether there are infinitely many elliptic curves with special properties, or if elliptic curves exhibit uniform behavior. It turns out that these properties are governed by the solutions to families of polynomial equations. In this award, the PI will study the solutions to these polynomial equations to deepen understanding of these uniformity questions. The PI will approach this by leveraging information from the geometry of the elliptic curves. The PI will also organize educational initiatives aimed at developing future talent in mathematics, as well as work to enhance digital infrastructure for mathematical research by contributing to mathematical databases. Quadratic Chabauty is a p-adic method for determining rational points on curves. It has proved to be a powerful tool in determining rational points on modular curves, which parametrize elliptic curves with a specified Galois image. A significant barrier to pushing the quadratic Chabauty method further is its reliance on a suitable plane model of the curve. The PI will determine rational points on certain families of modular curves by developing quadratic Chabauty algorithms that do not rely on explicit equations and instead leverage the modular nature of the curves. She will apply these approaches to determine rational points on interesting modular curves. This problem is motivated by conjectures such as Serre's uniformity question and Elkies' boundedness conjecture, which predict that, in certain families of modular curves, there are no rational points other than CM points or cusps. 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.