BC-MIT Number Theory Seminar
bis
Boston College, Maloney 560, Massachusetts
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. We also prove both conjectures for elliptic curves over function fields of characteristic p. Joint work with Kaiyuan Gu. ------------------------------------------------------------ Time: 4:30 - 5:30pm Speaker: Daniel Litt (University of Toronto) Title: The arithmetic of differential equations Abstract: In 1875 Fuchs asked for a criterion that determines when the solutions to an algebraic differential equation are themselves algebraic functions. Conjectural answers to this question have been given by Grothendieck and Katz, Ekedahl--Shepherd-Barron--Taylor, and Bost, in terms of the arithmetic of the differential equation's structure constants. I'll explain joint work in progress with Joshua Lam verifying these conjectures for a class of non-linear differential equations arising in algebraic geometry. This work is largely an excuse to try to better understand the arithmetic and geometry of algebraic differential equations and representations of fundamental groups of algebraic varieties.