The Arithmetic of Elliptic Curves

The Arithmetic of Elliptic Curves

3-4 hours of lectures per week.

Lecturer

Johan P. Hansen

Content

Elliptic curves are introduced from an algebraic geometric point of view. After a brief review of the basics of algebraic curves we proceed to the geometry of elliptic curves, the group law on elliptic curves and the determination of the ring of endomorphisms.

In particular we will study elliptic curves over finite fields and obtain the Hasse-Weil theorem on the number of rational points on an elliptic curves. The role of elliptic curves in cryptography will be treated.

Prerequisites

Algebra 1 and basic theory of algebraic geometry.

Literature

Joseph H. Silverman, The Arithmetic of Elliptic Curves, Springer, GTM, 1986.
Andreas Enge, Elliptic Curves and Their Applications to Cryptography: An Introduction, Kluwer Academic Publishers, 1999, see
http://www.lix.polytechnique.fr/Labo/Andreas.Enge/buch/buch/buch.html

Evaluation

The participants are supposed to perform a project (individual or in groups). The project shall contain a written and an oral presentation. Students who do not intend to take a degree in Mathematics or Statistics from the University of Aarhus, but wish to earn credits for a 2.dels course from the Department of Mathematics, should indicate at the beginning of the course that they wish to be examined.
The form of examination for these students will be active participation together with oral or written contributions.

Credits

10 ECTS

Quarter

Spring 2004