Introduction to p-divisible groups

The main aim of this thesis is to introduce a very important algebraic and geometric object: the notion of p-divisible group. p-divisible groups represent a profound intersection between the branch of algebraic geometry, number theory and representation theory, and provide a deep insight into the structure of algebraic varieties over fields of characteristic p. In order to give the definition of p-divisible groups and to analyze some of their main properties, we first organize a review of the main concepts of modern algebraic geometry, which stands at the base of objects such as p-divisible groups; we consider other important concepts such as the notions of group schemes, formal group schemes and formal groups in general, using a categorical framework. We proceed by analyzing the notion of p-divisible group given by Serre and Tate and comparing it with the one given later by Grothendieck, who renamed this groups Barsotti-Tate groups. Using important tools such as Witt vectors, Dieudonné modules and Cartiér duals, we show some properties of p-divisible groups and state some significant results. Finally, after introducing elliptic curves in the context of ableian varieties, we do the specific computations of the p-divisible group of those objects.