CAREER: Exceptional Points on Modular 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). Elliptic curves are among the most ubiquitous objects in modern number theory. They have far-reaching applications, both in theoretical mathematics – such as in the proof of Fermat's Last Theorem – and in information security where they form the basis of a cryptosystem commonly used to provide secure web browsing. The research in this project focuses on elliptic curves with unexpected arithmetic properties revealed by viewing these curves as distinguished points on a geometric object called a modular curve. In this context, the project will develop new tools for identifying these unusual elliptic curves, exploiting both the geometry of the modular curve and associated algebraic structures. In addition, the project includes several educational components, such as a training program in which master's degree students will serve as project leaders for undergraduates enrolled in a research exploration course. A central aim of the project is to broaden participation in the mathematical sciences, both at the undergraduate and graduate level.The main goal of this research is to explain isolated or sporadic points on modular curves, especially in the case where such points correspond to elliptic curves with a point (or a rational cyclic isogeny) of high order defined over a number field of unusually low degree. This is motivated by a desire to control the existence of such points in infinite families of modular curves, which lies at the heart of open questions raised by Mazur and Serre. A combination of tools will be employed, including geometric approaches stemming from Arakelov intersection theory and explicit computational techniques relating to Galois representations of elliptic curves. For certain modular curves, the project pursues an analogy between isolated points corresponding to elliptic curves with complex multiplication and those whose existence fails to be explained by any known geometric or modular phenomenon.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.