LEAPS-MPS: Isolated Points on Curves

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This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). The study of systems of polynomial equations has been a central theme in mathematics for thousands of years, and today related techniques have found applications in fields ranging from cryptography and computer science to mathematical biology. One approach to understanding the solution set of such a system of equations—which dates back to the Greek mathematician Diophantus—is to try to find only those solutions with coordinates in the integers or rational numbers. In cases where our polynomial equations define an algebraic curve, these rational solutions often sit inside a larger, finite set called the set of isolated points of the curve. These can be thought of as “unexpected” solutions to the system of equations, and the ultimate goal of this proposal is to develop new tools for identifying when these solutions occur. Several components of the investigation are suitable for graduate students and other early-career researchers, and their involvement constitutes one of the educational impacts of the proposed work. A second educational endeavor is a research training program involving both incoming Master’s degree students and early undergraduates. At all levels, this program will actively recruit students belonging to groups traditionally underrepresented in the sciences. The central aim of this proposal is to characterize isolated points on modular curves of genus at least 2, motivated by ties to several well-known classification problems and open conjectures. For a fixed family of modular curves, the work proposed falls into two main categories: (1) find all isolated points of any degree corresponding to a certain class of elliptic curves or (2) find all isolated points of a fixed degree corresponding to any elliptic curve. The project pursues several new explanations for isolated points, either by exploiting algebraic structures associated to the curve’s Jacobian or via Arakelov intersection theory. A specific aim is to apply these results to explain certain unexpected isogenies of elliptic curves over quadratic fields. In a different direction, the PI will study classes of isolated points associated to Q-curves motivated by ties to Serre’s Uniformity Conjecture for Galois representations attached to elliptic curves.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.