**Prerequisites**

Linear Algebra, groups, rings, fields. For example, the Leiden undergraduate courses Algebra 1, 2, 3. The notes for the Leiden Algebra courses can be found here: http://websites.math.leidenuniv.nl/algebra/ . For part of the course we will also need a small amount of complex analysis.

**Aim of the course**

Along various historical paths, the origins of elliptic curves can be traced back to calculus, complex analysis and algebraic geometry, and their arithmetic aspects have made them key objects in modern cryptography and in Wiles' proof of Fermat's last theorem. This course is an introduction to both the theoretical and the computational aspects of elliptic curves. The topics treated include a general discussion of elliptic curves and their group law, Diophantine equations in two variables, and Mordell's theorem. We will also discuss elliptic curves over finite fields with applications such as factoring integers, elliptic discrete logarithms, and cryptography.

**Rules about Homework / Exam**

The final grade will be based on a combination of a final exam and homework. Homework will count for 20% and the exam will count for 80%.

Each week there will be homework exercises, of which part will be marked to count towards the total homework mark. The homework should be handed in by the beginning of the following week's lecture, either on paper or electronically in this system.

**Literature**

We will mostly follow J.H. Silverman, "The arithmetic of elliptic curves" (SpringerLink). Students are expected to have this book (either electronically or on paper). Depending on your university's subscriptions, you may be able to get a pdf of this book at the given link, and if so, you can order it as a paperback book for EUR 25 from the publisher.

We may also refer to lecture notes: "Kernvak Algebra" by Peter Stevenhagen and Bart de Smit (PDF) and "Complex elliptic curves" by Peter Stevenhagen (PDF).

Other books on the topic include:

- J.H. Silverman and J. Tate, "Rational Points on Elliptic Curves" (SpringerLink)
- J.W.S. Cassels, "Lectures on Elliptic Curves" (Cambridge University Press)
- J. S. Milne, “Elliptic Curves”, available on the author's website.
- W. Fulton, "Algebraic Curves", also available online.
- J.H. Silverman, "Advanced topics in the arithmetic of elliptic curves" (SpringerLink)

- Docent: Martin Bright
- Docent: Marco Streng