Towards a classification of sporadic j-invariants

We say a point x on a curve C is sporadic if there exist only finitely many points on C of degree at most deg(x). In the case where C is the modular curve X_1(N), a non-cuspidal sporadic point can be thought of as corresponding to an elliptic curve with a point of order N defined over a number field of unusually low degree. Since every elliptic curve with complex multiplication gives rise to a sporadic point on X_1(N) for some positive integer N, we will primarily be interested in characterizing those non-CM elliptic curves which produce sporadic points. In this case, the problem is tied to several open questions in the arithmetic of elliptic curves, including Serre’s Uniformity Conjecture. In this talk, I will discuss recent results which aim to classify the elliptic curves producing sporadic points on X_1(N), spanning projects which are joint with David Gill, Jeremy Rouse, Lori Watson, and Filip Najman.