BC-MIT Number Theory Seminar
Boston College, Maloney 560, Boston, MA
Time: 3:00 - 4:00pm Speaker: Ziyang Gao (UCLA) Title: Towards uniformity in rational torsion and small points on abelian varieties Abstract: 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.…