Elliptic curves arise throughout number theory and arithmetic geometry, from the proof of Fermat’s last theorem to factoring algorithms and cryptography. After making the basic definitions, we will prove the fundamental theorems of Nagell-Lutz and Mordell-Weil (for a large class of elliptic curves), and then if time allows will discuss applications, for example to factoring integers.

Many of the most powerful tools for studying elliptic curves require fairly advanced knowledge of algebraic geometry. On the other hand, a large amount can be proven using only elementary techniques, and this will be the approach taken in this course – no background in algebraic geometry will be required. For those students with a good background in algebraic geometry, additional (optional) exercises will be provided to illustrate how these techniques can be applied.

**Number of hours**

2/week

**Literature**

Information on course literature will be provided in the webpage.

**Examination**

The course will be assessed through weekly homework (counting for 20%) and a final written exam (counting for the remaining 80%).

**Links**

Webpage Elliptic curves