Number Theory Seminar: Towards uniformity in rational torsion and small points on abelian varieties

On an elliptic curve E defined over a number field K, a celebrated theorem of Mazur, Kamienney, and Merel asserts the number of its rational torsion points is bounded above solely in terms of the degree of K — known as the uniform boundedness conjecture for elliptic curves. On the other hand, the Lang-Silverman conjecture predicts that the height of non-torsion rational points, divided by the Faltings height of E, is bounded below uniformly in terms of K.

In this talk, I will present a different method (inspired by Vojta’s proof of the Mordell Conjecture) to study both conjectures for abelian varieties of arbitrary dimension. Over number fields, we prove both conjectures assuming a suitable generalized version of Szpiro’s conjecture. Over function fields of characteristic 0, we resolve both conjectures unconditionally, giving a new proof of a remarkable result of Looper and Yap earlier this year — a common key ingredient is Deligne’s Arakelov Inequality. The bound obtained by our method is explicit and depends polynomially on the base field. We also prove both conjectures for elliptic curves over function fields of characteristic p.

Joint work with Kaiyuan Gu.