Extra Seminar - Dr. J. Sijsling (Ulm, Germany)

TITLE:            

Algebraic curves over number fields

ABSTRACT:

Algebraic curves over number fields are among the most fundamental objects of arithmetic geometry. Every polynomial equation in two variables whose coefficients are algebraic numbers defines such a curve, the study of which involves aspects of both geometry and number theory.

A measure of the complexity of an algebraic curve is its genus, which is nothing but the number of handles in this curve considered as a topological surface. Curves of genus 1 are known as elliptic curves, and played a crucial role in the proof of Fermat's Last Theorem. They are also used in practical applications involving chip cards and cryptography. Curves of higher genus are even more interesting from a theoretical point of view, while promising equally rewarding applications. Because of this combined interest, it is important to develop algorithms to make the theory of general curves, not just that of elliptic curves, accessible to computation and experimentation. This is the main focus of my research.