Sachi Hashimoto (Brown University)
Modular curves are, roughly speaking, curves whose points parameterize elliptic curves with extra structure on their torsion points. An open question in number theory is to find all rational points on all modular curves: this is known as "Mazur's program B". We will discuss some questions related to and inspired by Mazur's program B. In particular, we will discuss how to parameterize the set of all rational points on modular curves, conditional on a conjecture of Zywina. Using our parameterization we try to answer the following question: to what extent do the rational points on modular curves come from the intrinsic geometry of the curves? This is joint work with Maarten Derickx, Filip Najman, and Ari Shnidman.