Abbey Bourdon – Wake Forest University

Sporadic Points on Modular Curves

Our work is motivated by the following classification problem: For a fixed positive integer d, what finite groups arise as the torsion subgroup of an elliptic curve defined over a number field of degree d? In 1977, Mazur answered this question for elliptic curves over the rational numbers, and the classification for elliptic curves over quadratic fields was completed in 1992 through a series of papers by Kamienny, Kenku, and Momose. A few years later, Merel proved his celebrated Uniform Boundedness Theorem, which implies that if we fix d, then there are only finitely many groups that arise as the torsion subgroup of an elliptic curve defined over number field of degree d. However, the complete list of the groups that arise is unknown for any d>2.

A serious challenge in attempting to extend the classification is the need to identify all groups which arise for only finitely many isomorphism classes of elliptic curves–a phenomenon that does not occur for d=1 or d=2. The known examples correspond to elliptic curves with a rational point of order N appearing in unusually low degree; that is, they correspond to sporadic points on the modular curve X_1(N). In this talk, I will discuss recent results concerning sporadic points of X_1(N) which arise from elliptic curves with j-invariant of bounded degree. This is joint work with Ozlem Ejder, Yuan Liu, Frances Odumodu, and Bianca Viray.