Speaker：Lian Duan（ShanghaiTech University）

Time：2023-3-27, 15:00

Location：Conference Room 686 at the 6th floor of Shuli Building at Haiyun Campus

Abstract:

It is well known that over any number field, the Mordell Weil group of an abelian variety is finitely generated. Thus an abelian variety has only finitely many rational torsion points over a number field. This is in general not true if the base field is an infinite extension over Q. However, around 1981 Ribet proved that every abelian variety defined over a number field still has only finitely many torsion Q^ab-points, where Q^ab is the maximal abelian extension of Q. His result is then generalized by the works of Zarhin, Lombardo, and Rossler-Szamuely. In this talk, we will introduce another generalization of this result. That is, we will study the "torsion-finiteness" of an abelian variety over an infinite extension of the base field generated by adjoint all the torsion points of another abelian variety. Assuming the Mumford-Tate conjecture, we will give a criterion to the torsion-finiteness in terms of the Mumford-Tate groups of the related abelian varieties. In particular, when the conjecture is known, our theorem will deduce the unconditional results. This includes most cases of abelian varieties of dimension < ="3" and the cm cases. if time allows, we will also talk about the analogue over function field. this is a joint work with jeff achter, jiangxue fang, yuan ren and xiyuan wang.