**Dates:** Mondays, 14:00 (s.t.) in Seminarraum 6.

**Description:** The theory of elliptic curves is a very rich and broad subject with applications ranging from number theory to cryptography and dynamical systems. These curves are given by points \((x,y)\) in a field \(K\) satisfying an algebraic equation of the form \(y^{2}=x^{3}+Ax+B\).

Figure: The elliptic curve \(y^{2}=x^{3}-3x+3\) over the fields \(\mathbb{C}\), \(\mathbb{R}\) and \(\mathbb{F}_{101}\).

In this seminar we will present the theory of elliptic curves and explore some of its applications. We will start by introducing some basic aspects from algebraic geometry and number theory, define elliptic curves over different fields and study their properties, in particular their group structure. Once the foundations have been laid, a wide range of topics are available to be covered: rational points on elliptic curves (Mordell-Weil theorem), the congruent number problem (Tunnell's theorem), elliptic curve cryptography, moduli space of elliptic curves over \(\mathbb{C}\), modular forms and their applications, Fermat's Last Theorem,...

**Talks:**

- Introduction to algebraic geometry: notes, appendix
- Elliptic curves: notes
- Rational points on elliptic curves (Mordell’s Theorem): slides + notes
- Elliptic functions and elliptic curves over \(\mathbb{C}\): notes + work
- Moduli space of elliptic curves over \(\mathbb{C}\).
- Modular forms (I).
- Modular forms (II).
- Elliptic curve cryptography.

**Documents:**

- Announcement of the seminar.
- Summary of each of the talks.
- Programme of the course with references.