Title: Complexity of rational solutions to polynomial equations
Given a polynomial with rational coefficients, it is in general very hard to describe the set of solutions whose coordinates are rational numbers. The study of these rational solutions goes back over 2,000 years, and is the subject of some well-known theorems (eg. Fermat’s `last theorem’) and conjectures (eg. that of Birch and Swinnerton-Dyer). After giving an overview of a few known results and open questions, we will look more closely at the special case of elliptic curves. These are probably the most intensively studied of all rational solution sets, but remain in many ways very mysterious.
After giving the relevant definitions, we will describe certain functions (known as heights) which can be used to measure the arithmetic complexity of rational solutions on elliptic curves. We will introduce the notion of `locally decomposable' heights (more formally, height functions arising from Arakelov theory, though we will not use this language), which give a much better measure of complexity than other heights. We will apply these ideas to families of elliptic curves, and see some unexpected consequences of local decomposability.
Location: KdVI Meeting Room (Science Park 105-107, F3.20)