报告人：辜开源（北京大学）

时 间：2026年10月29日15:00

地 点：海韵园实验楼S106

内容摘要:

The Uniform Boundedness Conjecture (UBC) predicts that given an abelian variety A of dimension g defined over a number field K, the number of rational torsion points on A is bounded above solely in terms of g and K. While known for elliptic curves by the work of Mazur, Kamienny, and Merel, the conjecture remains widely open in higher dimensions. Recently, Looper and Yap proved its geometric analogue over function fields of characteristic zero, building on the method of Hindry--Silverman and Deligne's Arakelov inequality.

In this talk, I will present a new proof of the Geometric Uniform Boundedness Conjecture (GUBC), yielding explicit bounds with polynomial dependence on the base field. Our approach is inspired by Vojta's method of Diophantine approximation and incorporates new ideas from modern Arakelov geometry. The same method also give a proof of the Lang--Silverman conjecture over function fields of characteristic 0. Over number fields, we prove both conjectures conditionally on a higher-dimensional analogue of the Szpiro conjecture.

As a further application, our method yields bounds on the torsion subgroup of the Néron--Severi group of a smooth projective variety over C, a result that may be of independent interest.

This talk is based on a joint work with Ziyang Gao.  

个人简介：

辜开源是北京大学数学科学学院的在读博士，导师是袁新意教授。辜开源的研究领域是算术几何，在几何一致有界性猜想 (Geometric Uniform Boundedness Conjecture) 上已经取得了优秀的成果。

联系人：孙锐然