Mathematics & Statistics Departmental Research Seminar

Questions, answers and more questions about Mordell-Weil groups

Abstract:

An Abelian variety $A$ over a field $K$ is given by the set of solutions to a system of polynomial equations with coefficients in $K$ and that set can be equipped with a structure of an Abelian group. For instance, an elliptic curve $E/K$ will be (for simplicity) an Abelian variety given by an equation of the form $y^2 = x^3 + ax + b$, where $a, b \in K$ satisfy the condition $4a^3 + 27b^2 \neq 0$. The subset $A(K)$ of $A$ with solutions taken in a field $K$ has the Abelian group structure given by restricting the one of $A$. It is called the Mordell-Weil group of A and it plays a fundamental role in arithmetic geometry.

In this overview talk, I will discuss some questions that have been asked about the groups $A(K)$, partial and full answers to them and open conjectures. Recent results will also be presented, with a focus where $K$ is a function field of positive characteristic.