时间：2026年6月26日(星期五)，10:00-11:30
地点：E14-301
主讲人：Loïc Merel, Université Paris-Cité
报告主题：Finiteness of rational points: progresses and expectations on the asymptotics of uniformity
报告摘要：Faltings proved that the number of points (over a fixed number field $K$) of an algebraic curve of genus >1 is finite. But is the number of rational points on a curve of genus 2 over ${\bf Q}$ universally bounded?
Mordell and Weil proved that a set of $K$-rational points on an abelian variety (over a fixed number field $K$) is of finite type. But is there a uniform bound on the necessary number of generators for an elliptic curve over ${\bf Q}$?
Mazur proved that the torsion part of the group of rational points an elliptic curve over ${\bf Q}$ has order at most 16.
It is known that the torsion part of the group of K-rational points of an elliptic curve over a number field $K$ of degree $d$ over ${\bf Q}$ is bounded in terms of $d$ only. In particular, Oesterlé and Parent proved that the order is bounded by $(1+3^{d/2})^2$, and therefore exponential in $d$.
What is known and what is expected about the asymptotics of this theorem will be the main subject of this talk.
In particular, we will show how this relates to a variety of questions in arithmetic geometry and the theory of modular forms.
