**Prerequisites**

Linear algebra including vector spaces, solving linear equations, bases, and the Cayley-Hamilton theorem; group theory including the structure theorem for finitely generated abelian groups; ring theory: rings, ideals, polynomial rings, fields, and field extensions.

For example, the following chapters of the Leiden undergraduate courses Algebra 1, 2, 3 would be sufficient: 1–9, 11–14, 21, 22. The notes for the Leiden Algebra courses can be found here: http://websites.math.leidenuniv.nl/algebra/. The content of the Groningen undergraduate courses Group Theory and Algebraic Structures, together with chapters I, IV and V from the course Advanced Algebraic Structures would also suffice.

For part of the course we will also need a small amount of complex analysis: meromorphic functions, Cauchy's theorem, residues.

**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, with a focus on number-theoretical aspects. We find the structure of the elliptic curve group over various base fields such as the rationals, the complex numbers, and finite fields. We will also discuss applications such as factoring integers and cryptography.

- Docent: Steffen Muller
- Docent: Marco Streng