This is the git repository for my Master's thesis. I studied for it in Bordeaux, between January and June 2021, under the supervision of Professor Olivier Brinon. Its aim is to prove a crystalline comparison theorem for p-divisible groups, for a prime number p.
More specifically, let K be a complete discrete valuation field of characteristic 0 with perfect residue field of characteristic p. I have studied Messing’s crystalline Dieudonné theory and deformation theory of p-divisible groups, in order to prove the classification of p-divisible groups over the ring of integers of K with Breuil-Kisin modules. Using crystalline techniques, this allows to relate the p-adic Tate module of a p-divisible group and its Breuil-Kisin module by a period isomorphism, which in this case is nothing but the crystalline comparison isomorphism between the dual of the étale cohomology and that of the crystalline cohomology of the special fiber.
You can download the output pdf for the two versions I tagged in the appropriate section. You can also download the output of the last commit here.