## Lectures on Elliptic Curves

### HU Berlin, Winter 2019/20

### Schedule

Lectures | Thursday | 9:15 - 10:45 | Room 3.011 / RUD25 |

Thursday | 11:15 - 12:45 | Room 3.011 / RUD25 | |

Exercises | discussed during the lectures as needed |

There will be a written exam at the end of the course. You can choose between two dates:

Exam I | Thursday, 20/02/2020 | 11:00 - 13:00 | Room 1.013 / RUD25 |

Exam II | Thursday, 09/04/2020 | 11:00 - 13:00 | Room 1.013 / RUD25 |

### Problems/Notes

Part of the exam will be based on the weekly problem sheets. Please look at them carefully and do not hesitate to ask if you have any questions: The discussion of problems will be on demand, it is your task to keep the lecture interactive. The current sheets and some notes can be found here:

Problems 01 02 03 04 05 Lecture Notes (version 08/01/2020)

### Course topics

Elliptic curves are among the most fundamental objects of modern mathematics. In arithmetic geometry they have seen spectacular applications such as Andrew Wiles' proof of Fermat's last theorem, while in applied mathematics they are used for instance in cryptography.

The lecture will introduce the basic theory of elliptic curves, with a view towards some more advanced topics depending on the audience. We will begin with elliptic curves over $\mathbb{C}$, which are quotients of the complex plane by a lattice. Such quotients are objects of complex analysis, but they also admit an algebraic description as plane curves cut out by cubic polynomials. Passing to algebraic geometry, we can then consider elliptic curves over arbitrary fields. These are the simplest examples of abelian varieties: Projective varieties with an algebraic group structure. We will discuss their theory over finite fields and number fields, using ideas from complex analyis, algebraic geometry and number theory.

### Prerequisites

Basic algebra and complex analysis will be helpful. All other techniques can be developed on the way as needed, so the course can serve both as a self-contained introduction or as a complement to a course in algebraic/arithmetic geometry.

### Literature

- Cassels, J.W.S.,
*Lectures on Elliptic Curves,*LMS Student Series, CUP (1992) [www] - Husemöller, D.,
*Elliptic Curves,*Springer GTM (1987) [www] - Silverman, J.H.,
*The Arithmetic Theory of Elliptic Curves,*Springer GTM (1986) [www] - ---,
*Advanced Topics in the Arithmetic Theory of Elliptic Curves,*Springer GTM (1994) [www]