时间：2026年4月22日（星期三）14:00-16:00
地点：西湖大学云谷校区E14-116
主讲人：Evelina Viada, University of Göttingen
报告题目：Torsion and rational points of algebraic varieties
报告摘要：A leading principle in diophantine geometry is to describe via arithmetic properties large subsets of an algebraic variety $V$ that are non-dense in $V$.
A landmark of this principle is the Mordell Conjecture, proven by Faltings. It claims that a curve of genus at least 2 embedded in its Jacobian has a finite number of rational points. This beautiful result has fascinated many mathematicians, inspiring a number of articles. The proof of this theorem is not effective in the sense that we do not have an algorithm to determine such points. This is due to the fact that, in general, it is still unknown how to give an effective or an explicit bound for the height of such a set.
This remains a difficult open problem, in the context of the so-called effective or explicit Mordell conjecture.
I will present an overview of some explicit methods in the context of the Manin-Mumford and of the Mordell-Lang Conjectures. This allows us to explicitly determine the rational and torsion points on large families of curves.
I will first introduce the settings and basic geometric and arithmetic proprerties of algebraic group varieties. I will then present the central theorems and conjectures that we intend to investigate, such as the effective Manin-Mumford, the effective Mordell-Lang and the Torsion Anomalous Conjecture (=TAC).
Finally, I will explain how classical results of diophantine approximation and arithmetic geometry can be used in this context. To this aim I will present explicit examples and propose afordable new applications of our methods.
(Some of the results are in collaboration with Riccardo Pengo, and other with F. Veneziano)
