Elliptic curves are smooth projective curves of genus 1 with a marked point. Over a field of characteristic zero they are given by an equation of the form
$y^2 = x^3+ax+b$. They are at the boundary of our (conjectural) understanding of rational points on varieties and are subject of many famous conjectures as well as celebrated results. They play an important role in number theory.
The course will begin with an introduction to algebraic curves. We will then study elliptic curves over complex number, over finite fields, over local fields of characteristic zero and finally over number fields. Our goal will be to prove the Mordell-Weil theorem.