ELLIPTIC CURVES, THE ABC CONJECTURE, AND POINTS OF SMALL CANONICAL HEIGHT
(Notes From a Seminar Talk by Matt Baker)

William Stein

Abstract:

We begin by discussing the connection between the ABC conjecture and Szpiro's conjectural relationship between the conductor and discriminant of an elliptic curve. Then we discuss a conjecture of Lang which predicts, in particular, that among all elliptic curves $E$ over $\mathbb{Q}$ and all rational points $P$ in $E(\mathbb{Q})$, there is a point of smallest nonzero canonical height. After giving some computational examples due to William Stein, we will then discuss a theorem of Hindry and Silverman which implies that Lang's conjecture is implied by Szpiro's (and hence by the ABC) conjecture.