An elliptic curve over a field is a smooth projective curve of genus 1 which has a point defined over the basic field. In other words, a genus 1 curve is a curve which has a rational differential form without poles and zeros, or, equivalently, it is a smooth cubic curve in the two-dimensional projective space. On the set of points of elliptic curve which are defined over the basic field one can introduce the structure of an abelian group. If the ground field is the field of complex numbers, then this group is not very interesting, because it is isomorphic to a two-dimensional torus. If the ground field is finite, then one obtains a finite group which has a lot of applications in coding theory. If the ground field is the field of rational numbers, then one can prove that the resulting abelian group is finitely generated. A lot of famous conjectures in arithmetic algebraic geometry are connected with invariants of this group. It is supposed that participants of the course know the basics of Galois theory, the theory of p-adic numbers, and the theory of algebraic curves.

Financial support. The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, Agreement no. 075-15-2019-1614).

Lecturer
Osipov Denis Vasilievich

Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center