Elliptic curves are algebraic curves defined by a smooth cubic equation. A remarkable property of elliptic curves is the existence of a group law on its set of solutions. Algebraic geometry is a natural framework to study these elliptic curves. In this introductory course, we will begin by discussing classical results from algebraic geometry (like Hilbert’s Nullstellensatz), then we will study some particular cases of plane curves. Elliptic curves and the study of the structure of their groups of solutions will occupy a large part of the course.

For example elliptic curve over finite fields and cryptographic applications will be studied, as well a theorem of Hasse giving an estimation of the number of points of elliptic curves over finite fields. Concerning elliptic curves over the field of rational numbers, we will present a proof of the famous Morfell-Weil theorem as well as concrete examples where we can describe the group of rational points.

Here is a list of subjects which will be discussed in the course :

1. Basic commutative algebra, Hilbert Nullstellensatz, ideals of k[x,y], projective varieties, plane curves and intersection ;

2. Cubic equations and elliptic curves, group law, point counting over finite fields, Hasse theorem, cryptographic applications, elliptic curves over complex numbers, elliptic curves over Q and Mordell-Weil theorem.

**Prerequisites :** Basic knowledges in algebraic structures (groups, rings, modules, fields) and some elements of algebraic number theory. MAT451 and MAT552 could be useful.

**Langue du cours :** Français**Credits ECTS :** 5

- Teaching coordinator: Izquierdo Diego