Quadratic Chabauty is one of the premier methods for computing rational or integral points on curves in practice. In this talk, I will discuss unexpected failure modes of the method: when a certain heuristic predicts that the method should succeed in computing rational points, but it nonetheless fails. Focusing on the simplest interesting case -- integral points on rank 0 elliptic curves -- I will describe some new examples of failure cases of the method, along with some strong negative results to the effect that certain failure cases never occur. Ultimately, our results resemble a Mazur-style classification of torsion points on elliptic curves.

This is joint work in progress with Jennifer Balakrishnan.