Short description:
A smooth projective variety defined over a number field has its associated Hasse-Weil zeta function which encodes all of its arithmetic information. Fermat's last theorem was proved by Wiles et al. by showing that the Hasse-Weil zeta function of any elliptic curve defined over the rational numbers coincides with the L-function of a modular form of weight two. The million dollar worth Birch and Swinnerton-Dyer conjecture is concerned with the behavior of the zeroes of such Hasse-Weil zeta functions. More recently, and perhaps surprisingly, Hasse-Weil zeta functions of three dimensional Calabi-Yau manifolds have found applications in the study of black holes in the context of type II string theory.

In this course we will study a few examples where the Hasse-Weil zeta functions are well understood. Our list of examples will include the following:

- Elliptic curves with complex multiplication
- Modular curves
- Hyperelliptic curves over number fields
- Attractive K3 surfaces
- Del Pezzo surfaces over the rational numbers

We will first introduce all of the objects above over the field of complex numbers, so people interested solely in the complex geometry of these objects can benefit. After defining Hasse-Weil zeta functions in general, we will study these zeta functions for the above objects where a lot of their conjectural properties can be tested. To be more concrete, we will use the the computer algebra program MAGMA in which a number of algorithms have been implemented to construct these zeta functions explicitly. This will allow us to experiment and explore these zeta functions in a hands on way.

The course will consist of ten lectures, each one hour long. For students taking the course for credit, the final exam will be in the form of a thirty minute presentation on a topic of their choice.