BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Ziyang Gao (UCLA) DTSTART:20261110T200000Z DTEND:20261110T210000Z DTSTAMP:20261022T233906Z UID:BC-MIT/37 DESCRIPTION:Title: Towards uniformity in rational torsion and small points on abelian varieti es\nby Ziyang Gao (UCLA) as part of BC-MIT number theory seminar\n\nLe cture held in Maloney 560 at Boston College.\n\nAbstract\nOn an elliptic c urve E defined over a number field K\, a celebrated theorem of Mazur\, Kam ienney\, and Merel asserts the number of its rational torsion points is bo unded above solely in terms of the degree of K — known as the uniform bo undedness conjecture for elliptic curves. On the other hand\, the Lang-Sil verman conjecture predicts that the height of non-torsion rational points\ , divided by the Faltings height of E\, is bounded below uniformly in term s of K.\n\nIn this talk\, I will present a different method (inspired by V ojta's proof of the Mordell Conjecture) to study both conjectures for abel ian varieties of arbitrary dimension. Over number fields\, we prove both c onjectures assuming a suitable generalized version of Szpiro's conjecture. Over function fields of characteristic 0\, we resolve both conjectures un conditionally\, giving a new proof of a remarkable result of Looper and Ya p earlier this year — a common key ingredient is Deligne's Arakelov Ineq uality. The bound obtained by our method is explicit and depends polynomia lly on the base field. We also prove both conjectures for elliptic curves over function fields of characteristic p.\n\nJoint work with Kaiyuan Gu.\n LOCATION:https://researchseminars.org/talk/BC-MIT/37/ END:VEVENT END:VCALENDAR