Galoisdarstellungen und (phi,Gamma)-Moduln - Einzelansicht

We will further study the category of p-adic representations of the absolute Galois group G of a local field L which is a finite extension of Q_p. We will show that this category is equivalent in a natural way to the category of socalled etale (phi,Gamma)-modules. These are finitely generated modules over an explicit ring of Laurent series equipped with a semilinear operator phi and a semilinear action of a certain explicit group Gamma (in case L = Q_p the group Gamma simply is the group of units Z_p^*). In contrast to the Sen theory of the past semester no information is lost by passing to these modules.

But we gain that all the tools of semilinear algebra are available to understand Galois representations. In order to carry out this program we will learn tools like rings of Witt vectors, Lubin-Tate formal groups, and perfectoid fields.

Literature: P. Schneider, Galois Representations and (phi,Gamma)-Modules, Cambridge University Press 2017

The course assumes only basic knowledge about local fields from the first part of the course in the last semester. The Sen theory from the second part will play no role.