4.13 Eugen Hellmann, p-adic Hodge theory and deformations of Galois representations (part 5), 14 June 2019, INdAM Program

In this course we will introduce the deformation theory of Galois representations as well as p-adic Hodge theory with the aim of constructing and studying deformation spaces of Galois representations with a fixed p-adic Hodge type. In the rest part of the course we will discuss deformation theory with the focus on deformations of continuous representations of a profinite group - the interesting special case being the Galois group of a local or global field. In particular we will link obstruction classes and tangent vectors for deformations with Galois cohomology. In the second part of the course we want to study p-torsion and p-adic representations of the Galois group of a local p-adic field. This includes Fontaines equivalence of categories with so called etale (phi,Gamma)-modules, as well as an introduction to p-adic Hodge theory, which defines and studies certain interesting and important classes of p-adic Galois representation. We will finish this course with a discussion of potentially semi-stable deformation rings, i.e. deformation rings of local Galois representations with a prescribed p-adic Hodge theoretic behavior.