# p-adic Galois representations

**Lecture**

Monday, 2 - 4 pm, WSC-N-U-4.05

Wednesday, 2 - 4 pm, WSC-O-3.46

*first session on April 8*

**Problem session**

Tuesday, 2 - 4 pm, WSC-S-3.14

*first session on April 14*

**Prerequisites**

The lecture is independent of Algebraic Number Theory I & II. However, I do expect the participants to be well acquainted with the techniques provided by an introductory course on abstract algebra. This lecture forms part of the ALGANT-program for master students and will be given in English.

**Contents**

One of the main objectives of algebraic number theory is to understand the structure of the absolute Galois group of a local or a global field. The lecture will focus on absolute Galois groups of nonarchimedean local fields, i.e. of finite extensions of the p-adic numbers or of fields of Laurent series with coefficients in a finite field. The approach taken is to study a Galois group through its p-adic representations, i.e. through its continuous linear actions on finite dimensional p-adic vector spaces. The lecture will give a full account of the classification of p-adic Galois representations in terms of certain objects from semilinear algebra, the so-called étale φ- and (φ,Γ)-modules. I also plan to give an overview of the overconvergence theorem of Cherbonnier-Colmez. If time remains the course will end with an introduction to p-adic Hodge theory**.**

**Topics include**

Structure theory of local fields, p-adic representations of local Galois groups, étale φ-modules, fields of norms, étale (φ,Γ)-modules, overconvergence of p-adic Galois representations, p-adic period rings and the p-adic monodromy theorem

**References**

[1] L. Berger: An introduction to the theory of p-adic representations, Geometric aspects of Dwork theory. Vol. I, 255-292, Walter de Gruyter, 2004

[2] O. Brinon, B. Conrad: Notes on p-adic Hodge theory**, **notes from the CMI Summer School, preprint, 2009

[3 F. Cherbonnier, P. Colmez: Représentations p-adiques surconvergentes, Invent. Math. 133, 581-611, Springer, 1998

[4] J.-M. Fontaine, Y. Ouyang: Theory of p-adic Galois representations, preprint

[3] is a research article, whereas [2] and [4] are intended to become text books for graduate students. [1] gives an excellent overview of the subject and contains numerous additional references.

**Credits**

In order to be given credit points for this course you will need to take an oral exam in the end. In order to be admitted to the exam it is necessary to obtain at least 45% of the points on the problem sheets and to actively participate in the problem sessions.

**Problem sheets**

The problem sheets will be distributed and collected on Tuesdays during the problem sessions. You are encouraged to hand in your solutions in groups of up to three people.