Elliptic curves E can be given by plane projective cubic curves and so seem to be very simple objects. A first hint for more structure is that there is an algebraic addition law for the rational points. In fact, there is a natural isomorphism of E with its Jacobian variety, and so E is at the same time a curve of low degree and an abelian variety of smallest possible dimension. This is the reason for a very rich and deep theory behind making elliptic curves to ideal objects for both theoretical and experimental investigations, always with a strong algorithmic aspect. As outcome we find an abundance of key conjectures of arithmetic geometry inspired (and even proven) by elliptic curves.
It will be the purpose of the talk to explain some of these conjectures and results and, as important and rather astonishing side effect, state why these properties of elliptic curves make them to a most efficient and secure tool for public key crypto systems based on discrete logarithms.