The Mordell-Weil theorem and Chabauty's theorem

Abstract: Chapter 4 and Section 5.1 of Serre, Lectures on the Mordell-Weil theorem.

This talk will be split into two parts. In the first part, we will discuss the Mordell-Weil Theorem, which states that the abelian group of rational points on an abelian variety $A$ defined over a global field $K$ is finitely generated. We will show that this theorem follows from some classical finiteness results in algebraic number theory along with the theory of heights built up in previous talks. Time permitting, we will conclude the first part by proving a theorem of Neron which gives an asymptotic count for the number of points of bounded height on an abelian variety of rank $\rho$. In the second part, we will turn our attention towards curves of genus $g\ge2$. For such curves $C/K$, we will prove Chabauty's Theorem that $C(K)$ is finite if $\operatorname{rank}\operatorname{Jac}(C)(K) < g$ (finiteness of $C(K)$ is now known even when $C$'s Jacobian has large rank).

algebraic geometrynumber theory

Audience: advanced learners

STAGE

Series comments: STAGE (Seminar on Topics in Arithmetic, Geometry, Etc.) is a learning seminar in algebraic geometry and number theory, featuring speakers talking about work that is not their own. Talks will be at a level suitable for graduate students. Everyone is welcome.

Spring 2024 topic: The contents of Serre, Lectures on the Mordell-Weil theorem

Some topics might take more or less time than allotted. If a speaker runs out of time on a certain date, that speaker might be allowed to borrow some time on the next date. So the topics below might not line up exactly with the dates below.