Course on "Introduction to arithmetic of curves and surfaces", by Shehryar Sikanker

[Math Group]

COURSE ANNOUNCEMENT


Friday 7 April at 14:15

VENUE: Lecture room 136 (SISSA)

Shehryar Sikander (ICTP)


Title: INTRODUCTION TO ARITHMETIC OF CURVES AND SURFACES


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.


-- 
Koutou Mabilo
ICTP Mathematics Group
Leonardo Da Vinci Building
Strada Costiera no. 11
34151 Trieste, Italy

Tel. no.: +39-040-2240455
math at ictp.it
http://math.ictp.it