Joint IAS/PU Number Theory Seminar

An important open problem is to classify rational points on all modular curves, or equivalently to classify the possible adelic Galois images of elliptic curves over Q, as envisioned in Mazur’s Program B. However, this problem is inherently infinite in nature: there are infinitely many modular curves with rational points, including infinitely many with finitely many points and infinitely many with infinitely many points.

Building on Zywina’s notion of agreeable subgroups, we introduce the concepts of twist-parametrized and twist-isolated points on modular curves. Assuming standard conjectures on images of Galois representations of elliptic curves over Q, we show that all non-cuspidal, non-CM rational points on arbitrary modular curves are "twist-parametrized" by the rational points on 160 explicit modular curves.

This is joint work with Maarten Derickx, Sachi Hashimoto and Ari Shnidman.